.£55555 


'IN  MEMORIAM 

FLOR1AN  CAJOR1 


THE    PUBLIC    SCHOOL   ARITHMETIC 


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THE 


PUBLIC    SCHOOL    ARITHMETIC 


BASED   ON   McLELLAN    AND   DEWEY'S 

"PSYCHOLOGY    OF    NUMBER" 


BY 

J.    A.   McLELLAN,   A.M.,  LL.D. 

PRESIDENT  OF  THE  ONTARIO  NORMAL,  COLLEGE  ;   AUTHOR  (WITH  DR.  DEWEY) 

OF  "THE  PSYCHOLOGY  OF  NUMBER,"  "APPLIED  PSYCHOLOGY," 

"THE  TEACHER'S  HANDBOOK  OF  ALGEBRA,"  ETC. 

AND 

A.    F.    AMES,    A.B. 

HONOR   GRADUATE   IN   MATHEMATICS  ;    FORMERLY   MATHEMATICAL,   MASTEB 

ST.   THOMAS    COLLEGIATE    INSTITUTE,    ETC.  ;    SUPERINTENDENT 

OF    SCHOOLS,   RIVERSIDE,    ILL. 


f£0rk 
THE   MACMILLAN   COMPANY 

LONDON:  MACMILLAN  &  CO.,  LTD. 
1900 

All  rights  reserved 


COPYRIGHT,  1897, 
BY  THE  MACMILLAN   COMPANY. 


Set  up  and  electrotyped  June,  1897.      Reprinted  September, 
1897;  January,  July,  1898;  July,  1900. 


Nor  fa  not) 
J.  S.  Cushing  &  Co.  —  Berwick  &  Smith 
Norwood  Mass.  U.S.A. 


PREFACE 


THE  present  reaction  against  Arithmetic  as  a  school  study  is 
perhaps  due  partly  to  irrational  teaching  and  partly  to  the  grow- 
ing  tendency  to  make  things  interesting  to  the  learner  by  making 
them  easy.  This  reaction,  whatever  its  cause,  is  to  be  deplored. 
If  irrational  methods  of  teaching  have  led  to  waste  of  time  and 
unsatisfactory  results,  it  is  surely  the  part  of  wisdom  to  work, 
not  for  a  removal  of  the  subject  from  the  list  of  regular  studies, 
but  for  improvement  in  methods  of  teaching  it.  In  view  of  the 
present  cry  "  for  things,  not  abstractions,"  -  as  if  things  apart 
from  abstractions,  the  mind's  action  upon  things,  Avere  the  only 
reality,  -  -  there  is  urgent  need  of  giving  Arithmetic  its  rightful 
place  and  having  it  taught  on  sound  psychological  principles. 
When  so  taught,  no  subject  can  be  substituted  for  this  "Logic  of 
the  Public  Schools."  There  has  been  much  nonsense  uttered  by 
those  who  are  clamorous  for  "  Content ''  studies  as  opposed  to 
"  Form "  studies.  How  is  it  possible  to  separate  Form  and 
Content,  and  regard  the  one  as  good  in  itself  and  the  other 
as,  at  best,  a  necessary  evil  ? 

"  In  the  case  of  Number,  form  represents  the  measured  adjust- 
ment of  means  to  an  end,  the  rhythmical  balancing  of  parts  in  a 
whole,  and  therefore  the  mastery  of  form  represents  directness, 
accuracy,  and  economy  of  perception,  the  power  to  discriminate 
the  relevant  from  the  irrelevant,  and  ability  to  mass  and  converge 
relevant  material  upon  a  destined  end ;  represents,  in  short,  pre- 
cisely what  we  understand  by  good  sense,  by  good  judgment,  the 


vi  PREFACE 

power  to  put  two  and  two  together.  When  taught  as  this  sort  of 
form,  Arithmetic  affords,  in  its  own  place,  an  unrivalled  means  of 
mental  discipline."  * 

To  help  both  teacher  and  pupil  to  make  the  most  of  this 
" unrivalled  means  of  mental  discipline"  is  the  purpose  of  "The 
Public  School  Arithmetic."  The  treatment  of  the  subject  is  in 
line  with  what  is  believed  to  be  the  true  idea  of  number  and 
numerical  operations  as  developed  in  McLellan  and  Dewey's 
"Psychology  of  Number,"  —a  work  which,  to  use  the  words  of 
one  of  our  greatest  educators,  "  has  caused  more  stir  among 
teachers,  and  elicited  a  greater  number  of  favorable  opinions 
than  any  other  pedagogical  book  of  the  day."  Indeed,  "The 
Public  School  Arithmetic'1  has  been  prepared  in  response  to 
requests  from  teachers  and  educators  of  every  grade,  for  a  text- 
book on  Arithmetic  based  on  the  principles  set  forth  in  "The 
Psychology  of  Number."  Some  of  the  special  features  of  the 
book  may  be  noted : 

1.  The  treatment  of  the  subject  is  in  strict  line  with  the  idea 
of  number  as  measurement  —  a  process,  that  is,  by  which  the  mind 
makes  a  vague  whole  of  quantity  definite ;  the  mental  sequence 
being  the  undefined  whole,  the  units   of  measure,   the  defined 
whole.     The  book  from  the  beginning  stimulates  and  promotes 
this  spontaneous  action  of  the  mind.     For  rational  method  in 
arithmetic,  i.e.  method  in  harmony  with  psychical  development, 
a  true  conception  of  number  is  a  prime  necessity ;  a  false  idea  of 
number  hinders  the  normal  action  of  the  mind,  dulls  perception 
and  reason,  cultivates  habits  of  inaccurate  and  disconnected  atten- 
tion; in  a  word,  makes  the  subject  all  but  worthless  as  a  means 
of  mental  discipline. 

2.  This  true  idea  of  number  running  through  the  whole  work 
establishes  the  unity  of  the  whole.     The  unity  of  arithmetic  is 
the  unity  of  the  fundamental  operations,  and  is  the  indispensable 
condition  of  interest  in  the  study.     The  book  practically  develops 

*  The  Psychology  of  Number,  preface,  page  xii.  . 


PREFACE  vii 

and  applies  the  true  relation  between  addition  and  subtraction 
(not  "two"  rules),  between  addition  and  multiplication  (not 
"  identical '  operations),  between  multiplication  and  division 
(psychologically  not  two  —  much  less  four  —  processes),  and 
between  these  operations  and  fractions.  This  leads  to  the  true 
idea  of  number,  gives  meaning  to  the  primary  operations,  culti- 
vates facility  in  their  applications,  and  imparts  to  all  operations 
and  processes  a  vitalizing  interest. 

3.  The  real  meaning  of  the  primary  operations  being  practically 
developed,  fractions  are  divested  of  their  traditional  difficulty  by 
being  placed  in  their  true  relation  to  "  integers  " ;  requiring,  in 
fact,  merely  the  conscious  recognition  of  ideas  which  have  from 
the  first  been  freely  used.     The  ideas  of  multiplication  and  divi- 
sion, which  are  implicit  in  all  the  fundamental  operations, -- in 
the  very  idea  of  number  —  become  explicit  in  fractions.     Every 
"  rule  "  in  fractions  has  a  meaning,  and  therefore  has  an  interest 
that  leads  to  facile  mastery. 

4.  The  same  remark  applies  to   Greatest  Common  Measure, 
Least  Common  Multiple,  etc.,  the  real  significance  of  which  is 
rarely  comprehended  by  the  pupil.     The  clear  conception  of  what 
number  is,  gives  clearness  and  interest  to   every  principle    and 
process.     This  is  true  even  of  the  "mechanical"  operations. 

5.  As  fractions  are  but  an  application  of  "  The  Simple  Rules," 
especially  multiplication  and  division  (ratio),  so  percentage  in  all 
its  forms  is  but  an  application  of  fractions.     The  pupil   is  not 
perplexed  with  numerous  "  rules  and  cases  "  ;  he  grasps  the  idea 
of  number  as  measurement,  and  all  the  "  rules  "  referred  to  simply 
require  the  use  of  the  one  underlying  principle.     In  fact,  the 
pupil  is  constantly  passing  from  ideas  to  principles  and  "  rules," 
which  he  formulates  for  himself. 

6.  Great  care   has  been  taken  in  selecting  and  grading  the 
examples,  many  of  which  have  been  prepared  specially  for  this 
work.     The  typical  solutions  are  clear  and  concise;  there  is  no 
perplexing  verbiage  under  the  name  of  "  logical  analysis." 


viii  PREFACE 

7.  This  treatment  of  arithmetic  will  prove  a  good  preparation 
for  algebra.     The  main  difficulty  for  the  beginner  in  algebra  is  in 
the  symbols.     He  must  learn  to  think  in  the  language  of  algebra ; 
a  higher  degree  of  abstraction  is  required.     If,  then,  the  symbols 
and  operations  in  arithmetic  have  little  meaning  for  him,  what  is 
his  plight  when  he  is  confronted  with  the  higher  abstractions  of 
algebra  ? 

He  passes  from  abstractions  which  have  little  meaning  to 
abstractions  which  have  no  meaning.  The  teacher  is  obliged  to 
lay  a  foundation  which  should  have  been  laid  in  arithmetic.  Too 
often  this  foundation  is  never  laid ;  essential  ideas  are  left  vague, 
and  all  advanced  work  shares  in  the  vagiieness.  For  the  highest 
abstractions  in  mathematics  are  an  evolution  from  the  lowest. 
On  the  other  hand,  to  the  student  who  lias  grasped  the  true  mean- 
ing of  number  and  arithmetical  operations,  the  transition-  -hardly 
a  transition  —  is  easy  and  sure.  There  is  no  unbuilding  and  re- 
building to  be  done.  Algebra  becomes  a  thing  of  beauty  and  of 
power,  an  effective  factor  in  "  unfolding  the  Laws  of  the  Human 
Intelligence." 

8.  The  idea  and  method  of  the  book  has  been  tested  in  actual 
work ;  and  if  interest,  not  to  say  enthusiasm,  on  the  part  of  both 
teachers  and  pupils  is  any  test,   there  is   good  reason  for  the 
appearance  of  "  The  Public  School  Arithmetic." 

9.  This  book  will  be  followed  in  a  short  time  by  a  Primary 
book  in  which  the  measuring  idea  will  be   strictly  carried  out. 
Meantime  every  method  and  every  device  given  in  the  Primary 
book  is  being  tested  in  actual  work. 

We  are  indebted  to  J.  C.  Glashan's  Arithmetic  for  some  good 
suggestions  and  problems. 

It  is  recommended  that  starred  chapters  or  parts  of  chapters  be 
omitted  from  the  Grammar  School  Course. 

JUNE,  1897. 


CONTENTS 

CHAPTER   I 

PAGB 

DEFINITIONS 1 

CHAPTER   II 
NUMERATION  AND  NOTATION 10 

CHAPTER   IH 
ADDITION 20 

CHAPTER   IV 
SUBTRACTION      ...........       32 

CHAPTER   V 
MULTIPLICATION 43 

CHAPTER   VI 
DIVISION 59 

CHAPTER   VTI 
COMPARISON  OF  NUMBERS 76 

CHAPTER   VIII 
SQUARE  ROOT 80 

CHAPTER  IX 

GREATEST  COMMON  MEASURE  AND  LEAST  COMMON  MULTIPLE  .       87 

ix 


x  CONTENTS 

CHAPTER  X 

PAGE 

FRACTIONS 100 

CHAPTER  XI 
DECIMALS 139 

CHAPTER  XII 
COMPOUND  QUANTITIES 157 

CHAPTER  XIII 
PERCENTAGE 197 

CHAPTER  XIV 
INTEREST 239 

CHAPTER  XV 
RATIO  AND  PROPORTION 274 

CHAPTER  XVI 
POWERS  AND  ROOTS  ..........     288 

CHAPTER  XVII 
MENSURATION 297 

CHAPTER  XVIII 

METRIC  SYSTEM 314 

CHAPTER  XIX 
MISCELLANEOUS  EXERCISE  ...  .     322 


ARITHMETIC 


CHAPTER   I 

DEFINITIONS 

1.  Arithmetic  is  the  science  of  numbers  and  the  art  of  com- 
puting by  them. 

2.  A  Unit  is  a  quantity  regarded  as  a  standard  of  reference 
with  which  to  compare  or  measure  quantity  of  the  same  kind; 
as  $1,  1  five-dollar  bill,   1  in.,   1  lb.,  1  two-ounce  weight, 
1  three-inch  measure,  1  doz.  eggs. 

3.  Number  is  the  repetition  of  the  unit  to  measure  a  given 
quantity,  i.e.  number  determines  the  hoiv  much  of  the  quan- 
tity by  determining  hoiv  many  units  make  up  the  quantity. 

4.  The  number  by  itself  shows  the  relative  value  or  the 
Ratio  of  the  quantity  to  the  unit.      Thus   the  ratio   of  the 
quantity  $6  to  the  unit  $1  is  the  number  6;    the  ratio  of 
the  quantity  4  (3  apples)  to  the  unit  8  apples  is  the  num- 
ber 4.     The  number  and  the  unit  together  show  the  absolute 
magnitude  of  the  quantity.     If  the  quantity  is  measured  by 
the  number  8  and  the  unit  of  weight  4  lb.,  then  its  absolute 
magnitude  is  8  times  4  lb.  or  32  lb. 

If  I  measure  the  length  of  a  room  with  a  two-foot  measure 
and  find  it  to  be  16  ft.,  the  unit  of  length,  2  ft.,  has  been 


2  ARITHMETIC 

repeated  8  times  to  measure  the  length  of  the  room,  and 
the  ratio  of  the  length  of  the  room  to  the  unit  of  measure  is 
represented  by  the  number  8. 

If  a  box  has  been  found  to  contain  30  doz.  of  eggs,  the 
unit,  1  doz.  eggs,  has  been  counted  30  times  to  measure  the 
quantity  of  eggs. 

If  I  buy  3  qt.  of  milk,  the  milkman  fills  and  empties  the 
unit  of  capacity,  his  one-quart  measure,  3  times  in  order  to 
measure  the  quantity  of  milk. 

5.  In  the  above  instances  the  wholes  to  be  measured  are : 
the  length  of  the  room,  the  eggs  in  the  box,  and  the  milk 
sold.     The  units  of  measure  are  2  ft.,  1  doz.,  and  1  qt.     The 
numbers  telling  how  many  of  these  units   are   needed   are 
8,  30,  and  3. 

The  unit  of  measure  may  itself  be,  and  in  exact  computa- 
tion is,  measured  or  compared  with  some  other  unit,  for  con- 
venience called  the  primary  unit ;  that  is,  it  may  be  a  part  or 
a  multiple  of  such  unit. 

6.  $  1,  1  in.,  1  lb.,  are  all  primary  units.     A  five-dollar 
bill,  a  two-pound  weight,  a  four-inch  measure,  are  examples 
of  derived  units,  because  the  primary  units,  $  1,  1  lb.,  1  in., 
are  repeated  5,  2,  and  4  times  respectively,  to  give  the  derived 
units. 

If  a  quantity  of  chestnuts  be  counted  into  groups  of  4  or 
5,  the  derived  units,  4  or  5  chestnuts,  and  the  number  of 
these  groups  or  derived  units  measures  the  whole  quantity 
of  chestnuts.  Similarly,  eggs  counted  by  4's  and  6's,  and 
apples  by  3's  or  5's,  are  examples  of  the  use  of  the  derived 
unit.  The  primary  units  are  1  egg  and  1  apple,  while  4  eggs, 
6  eggs,  3  apples,  and  5  apples  are  derived  units. 


DEFINITIONS  3 

7.  With  reference  to  1  ct.,  $  1  is  a  derived  unit,  and  100 
is  the  number  expressing  $1  in  terms  of  the  primary  unit, 
1  ct. 

Similarly,  1  wk.  is  a  derived  unit,  and  7  is  the  number 
expressing  1  wk.  in  terms  of  the  primary  unit,  1  da. 

In  ^  ft.,  the  primary  unit  of  reference  is  1  ft.  The  foot 
is  divided  into  4  equal  parts,  one  of  which  is  the  derived  unit 
of  measure.  The  number  3  shows  how  many  of  these  de- 
rived units  make  up  the  given  length. 

Exercise  1 

1.  Name  three  units  of  length  used  to  measure  short  distances, 
and  state  the  number  of  times  each  unit  must  be  repeated  to  make 
the  next  larger. 

2.  What  unit  of  length  is  used  in  stating  the  distance  between 
two  cities  ? 

3.  Name  instances  in  which  1  sec.  is  used  as  the  unit  of  time. 
1  min.     1  hr.    1  da.    1  mo.    1  yr.     State  how  often  each  of  these 
units  must  be  repeated  to  make  the  next  larger. 

4.  What  is  the  prime  standard  unit  for  money  value?     For 
weight  ?     For  area  ?     For  length  ?     For  time  ?     For  volume  ? 

5.  What  unit  of  area  is  used  to  convey  a  definite  idea  of  the 
size  of  a  farm  ?     Of  a  country  ? 

6.  What  unit  is  used  to  measure  wood  ? 

7.  With  what  unit  of  capacity  is  milk  measured  ?    Kerosene  ? 

8.  What  unit  is  used  to  measure  a  quantity  of  strawberries? 
Potatoes  ?     Why  are  these  convenient  units  for  the  purpose  ? 

9.  State  at  least  three  reasons  why  the  bushel  would  be  an 
inconvenient  unit  to  measure  strawberries. 

10.  Early  in  the  season  strawberries  are  sold  in  pint  boxes; 
later,  in  quart  boxes.  Explain  why  different  units  of  capacity 
are  chosen. 


4  ARITHMETIC 

11.  Why  is  the  pint  box  chosen  as  the  unit  to  measure  red 
raspberries  in  preference  to  the  quart  ? 

12.  Name  different  quantities  which  are  weighed  and  sold  by 
the  Ib.     By  the  oz.     By  the  T. 

13.  Give  instances   in  which   the   following   units   are   used: 
1  sheet,  1  quire,  1  doz. 

14.  In  each  of  the  following   quantities  name  the  units  and 
give  the  ratio  of  each  quantity  to  its  primary  unit:  5  ft.,  4  hr., 
6  sq.  in.,  7  qt.,  365  da.,  12  oz. 

15.  What  is  the   quantity  which  contains  the  unit  4  times, 
when  the  unit  is  6  in.  ?     9  hr.  ?     8  yr.  ? 

16.  If  the  unit  is  $4,  and  this  unit  is  repeated  6  times,  what 
quantity  will  be  produced  ? 

17.  If  the  ratio  of  the  size  of  a  farm  to  the  unit  of  area,  8  A., 
is  equal  to  6,  what  is  the  size  of  the  farm  ? 

18.  In  the  following  examples,  what  are  the  quantities  which 
contain  their  respective  units  the  given  number  of  times  ? 

UNITS  OF  MEASUBE  NUMBERS 

$2  6 

8  qt.  2 

7  da.  3 

5  hr.  4 

1  doz.  16 

4  in.  4 

19.  Name  the  primary  units  of  measure,  name  in  two  ways 
the  derived  units,  and  state  the  number  of  derived  units  which 
measure  these  quantities :  $  T?Q-,  £  ft.,  -§•  yd.,  J-  of  a  dime,  -f-  of  a 
wk.,  |  of  a  da.,  f  of  a  doz.  eggs. 

20.  Name  the  coin  which  gives  the  derived  unit  of  value  in 
each  of  the  following :  $  ft,  $  £,  $  £J,  f  f f,  $  10,  $  5,  $  20,  f  of  a 
nickel,  A-  dime,  %  of  a  quarter  of  a  dollar,  f  of  half  a  dollar,  f  of 
half  a  dollar,  ff  of  half  a  dollar, 


DEFINITIONS  5 

21.  The   following   quantities  contain  their   respective   units 
how  often  ? 

QUANTITY  UNIT 

21  da.  7  da. 

24  hr.  8  hr. 

1  gal.  1  qt. 

1  min.  1  sec. 

10  dimes  2  dimes 

$  18  worth  of  hats                    $3  for  1  hat 

30  ct.  worth  of  rnilk  6  ct.  a  qt. 

22.  What  is  the  unit  of  measure  in  reckoning   population? 
How  many  of  these  units  give  the   population  of  the  town  in 
which  you  live  ? 

23.  What  is  the  number  of  times  each  of  the  following  units 
must   be  repeated   to  make  the  next   higher  unit :    1  in.,  1  ft., 
1  ct.,  1  dime,  1  da.,  1  hr.,  1  qt.  ? 

24.  How  many  times  must  the  following  units  of  measure  be 
repeated  to  make  3  ft. :  2  in.,  3  in.,  4  in.,  6  in.,  9  in.,  12  in.  ? 

25.  State  how  each  of  the  following  units  may  be  derived  from 
the  next  higher:  1  ft.,  1  in.,  50  ct.,  25  ct.,  1  da.,  1  min.,  1  qt., 
and  1  pt. 

26.  A  quantity  of  cherries  is  measured  by  using  as  the  unit 
as  many  cherries  as  will  fill  a  dish  holding  3  qt. ;   9  of  these 
dishes    are    filled.      How   many    qt.    are    there    in    the    whole 
quantity  ? 

27.  At  30  bu.  to  the  A.,  how  many  bu,  would  there   be  on 
10  A.  ?     What  is  the  unit  here  ?     What  gives  this   particular 
unit  ?     If  10  bu.  to  the  A.,  how  many  A.  to  produce  an  equal 
quantity  ? 

28.  What  coin  is  equal  in  value  to  10  of  the  unit  1  ct.  ?     100 
of  the  unit  1  ct.  ?     10  of  the  unit  1  dime  ? 

29.  What  coin  is  equal  to  10  of  the  unit  1  nickel?    5  of  the 
unit  1  nickel  ?     20  of  the  same  unit  ? 


tf  ARITHMETIC 

30.  How  many  ct.  are  there  in  5  of  the  unit  $1?     6  of  the 
unit  $  1  ?     10  of  the  unit  $  1  ? 

31.  What  coin  is  equal  in  value  to  500  of  the  unit  1  ct.  ? 
100  of  the  unit  1  nickel  ? 

32.  What  coin  is  equal  to  one-tenth  of  a  five-dollar  gold  piece  ? 


8.  In  the  preceding  diagram  we  have  36  dots,  signifying 
36  units  of  any  kind,  arranged  in  4  rows  of  9  dots  each,  and  at 
the  same  time  9  rows  of  4  dots  each.  Hence  we  think  of  36 
as  equal  to  4  times  9  or  9  times  4.  Arrange  the  dots  to  show 
that  36  is  equal  to  3  x  12  or  12  x  3,  and  also  to  2  x  18  or  18  x  2. 

4  and  9  are  called  factors  of  36,  and  36  is  called  the 
product  of  4  and  9. 

This  illustrates  a  law  of  great  importance  in  Arithmetic. 

Thus  we  think  of  24  as  equal  to  2  x  12  or  12  x  2,  3  x  8  or 
8  x  3,  4  x  6  or  6  x  4. 


9.  If  in  the  above  arrangement  we  think  of  each  dot  as  rep- 
resenting $1,  then  the  diagram  shows  that  $12  -=-  $2  =  6. 

What  other  measurement  is  shown  by  the  same  arrange- 
ment ? 


Exercise  2 


1.  Arrange  30  dots  in  rows  in  as  many  ways  as  you  can,  and 
express  the  results  as  in  the  preceding  paragraph. 

2.  Express  the  following  numbers  as  products  in  two  ways: 
6  (i.e.  2  x  3  or  3  x  2),  8,  10,  15,  21,  and  35. 


DEFINITIONS  7 

3.  Express  the  following  numbers  as  products  in  as  many  ways 
as  possible,  and  arrange  the  products  in  corresponding  pairs :  12, 
16,  20,  28,  42,  and  60. 

4.  Give  all  the  factors  of  32,  40,  and  48. 

5.  Place  dots  to  show  the  measurement  of  $  20  by  a  $  5  unit. 
What  other  measurement  does  it  show  ? 

6.  Place  dots  to  show  the  measurement  of  $24  by  a  $2  unit; 
a  $  3  unit ;  a  $  4  unit.     AVhat  other  measurements  are  shown  ? 

7.  What  is  the  price  of  6  yd.  of  cheese-cloth  at  8  ct.  a  yd.? 
Explain  each  of  these  statements  : 

The  whole  price  =  6  (8  ct.). 
The  whole  price  =  8  (6  ct.). 

8.  Show  that  10  units  of  $  5  each  is  equal  to  5  units  of  $  10 
each. 

9.  If  a  line  3  ft.  long  is  repeated  6  times  to  measure  the 
length  of  a  room,  how  long  is  the  room  ?     How  often  would  a 
line  6  ft.  long  have  to  be  repeated  to  measure  the  room  ? 

10.  How  many  apples  will  be  required  to  make  7  rows  with 
9  apples  in  each  row  ?    What  is  the  unit  of  measurement  ?    What 
other  convenient  unit  might  be  used  ?     What  would  be  the  ratio 
of  the  whole  quantity  to  this  unit  ? 

11.  Draw   a   straight   line   36   in.   long,    cut   strips    of   paper 
respectively  6  in.,  7  in.,  8  in.,  9  in.,  10  in.,  11  in.,  and  12  in. 
long.     Measure  along  the  line  3  times  with  each  of  these  strips 
of  paper. 

Use  a  yardstick  divided  into  in.  to  measure  your  results,  and 
prove  the  following : 

3  x  6  in.  =  18  in. ;  3  x    9  in.  =  27  in. ; 

3  x  7  in.  =  21  in. ;  3  x  10  in.  =  30  in. ; 

3  x  8  in.  =  24  in. ;  3  x  11  in.  =  33  in. ; 

3  x  12  in.  =  36  in. 


g  ARITHMETIC 

12.  Draw  a  line  36  in.  long.     Make  a  3-in.  measure.     Measure 
along  the  line  respectively  6,  7,  8,  9,  10,  11,  and  12  times,  and 
prove  that : 

6  X  3  in.  =  18  in. ;     9x3  in.  =  27  in. ; 

7x3  in.  =  21  in. ;  10  x  3  in.  =  30  in. ; 

8x3  in.  =  24  in. ;  11  x  3  in.  =  33  in. ; 

12  x  3  in.  =  36  in. 

13.  What  quantities   are   measured   by  the   following:   6x4 
in.,  4x6  in.,  5  x  $  8,  8  x  $5,  10  x  5  pears,  5  x  10  pears  ? 

14.  What  two  units  of  length  each  longer  than  4  in.  can  be 
used  to  measure  a  line  35  in.  long  ?     State  in  each  case  the  ratio 
of  the  length  of  the  whole  line  to  each  unit. 

15.  What  are  the  convenient  units  of  money  to  pay  a  debt 
of  $35?     |80? 

16.  What  are  convenient  units  to  pay  debts  of  75^  ? 
34^  ?     87^  ? 

17.  A   fruit    dealer    sells    apples    at    the    rate   of    3   for 
What  is  the  unit  to  measure  his  apples  ?     What  is  the  unit  of 
value  ?     How  many  units  are  there  in  24  apples  ?     What  is  their 
selling  price  ? 

18.  Bananas  are  sold  at  the  rate  of  4  for  5^.     What  is  the 
measuring  unit  for  the  bananas  ?     What  is  the  unit  of  value  ? 
How  many  units  of  value  in  20  bananas  ?     36  bananas  ?     What 
is  their  value  ? 

19.  Oranges  are  sold  at  20^  a  doz.     What  are  the  two  meas- 
uring units  ? 

20.  If  A  can  do  a  piece  of  work  which  is  represented  by  36 
units  in  9  da.,  how  much  will  he  do  in  1  da.  ? 

21.  If  a  piece  of  work  is  represented  by  60  units  and  A  can  do 
5  units  in  one  da.,  in  how  many  da.  can  he  do  the  entire  work  ? 

A          B  o  D          E  F 

I i | i i i 

22.  AB  is  a  line  which  represents  any  primary  unit  of  measure, 
and  AC,  AD,  AE,  and  AF  are  derived  units.     What  part  is  the 


DEFINITIONS  9 

primary  unit  AB  of  each  of  the  derived  units  ?  What  is  the 
ratio  of  each  derived  unit  to  the  primary  unit  ?  If  AF  is  the  pri- 
mary unit,  what  would  AB  be  ?  What  is  the  ratio  of  the  derived 
unit  AF  to  the  derived  unit  AD  ?  Of  AD  to  AF?  Of  AC  to 
AE?  OfAEtoAC?  Of  AF  to  AC?  Of  AC  to  AF? 

23.  One  field  contains  7  units  of  area,  and  a  second  field  con- 
tains 9.     What  is  the  ratio  of  the  area  of  the  first  field  to  the 
second  ?   Of  the  second  field  to  the  first  ?   Illustrate  by  a  drawing. 

24.  The  distance  from  A  to  B  is  divided  into  5  parts  of  3 
mi.  each,  and  that  from  A  to  C  into  6  parts  of  3  mi.  each.     What 
is  the  ratio  of  the  distance  AB  to  AC?     Of  AC  to  AB?     Illus- 
trate your  answer  by  a  diagram. 

25.  The  money  in  my  purse  is  measured  by  the  number  8  and 
the  unit  $5.     I  owe  a  debt  measured  by  the  number  6  and  the 
unit  $  5.     What  is  the  ratio  of  the  debt  to  the  money  in  my 
purse  ?    What  is  the  ratio  of  the  money  in  my  purse  to  the  debt  ? 
How  much  shall  I  have  left  after  paying  my  debt  ? 

26.  If  the  amount  of  work  required  to  dig  a  trench  800  yd. 
long  is  represented  by  40  units,  what  does  1  unit  represent  ? 


CHAPTER   II 

NUMEKATION  AND  NOTATION 

10.  Numeration  is  counting,  or  the  expression  of  number 
in  words. 

The  ordinary  system  of  numeration  is  the  Decimal  System, 
so  called  because  it  is  based  on  the  number  ten. 

11.  The  names  of  the  first  group  of  numbers  in  regular 
succession  are:  one,  two,  three,  four,  five,  six,  seven,  eight, 
nine. 

Other  number-names  are :  ten,  hundred,  thousand,  million, 
billion,  trillion,  etc. 

12.  The  number  one  applied  to  any  unit  denotes  a  quantity 
which  consists  of  a  single  unit  of  the  kind  named. 

The  number  two  applied  to  any  unit  denotes  a  quantity 
which  consists  of  one  such  unit  and  one  unit  more. 

The  number  three  applied  to  any  unit  denotes  a  quantity 
which  consists  of  two  such  units  and  one  unit  more. 

And  so  on  with  the  numbers  four,  five,  six,  seven,  eight, 
nine ;  applied  to  any  unit  they  denote  quantities  increasing 
regularly  by  one  such  unit  with  each  successive  number. 

13.  The  number  next  following  nine  is  ten,  which  applied 
to  any  unit  denotes  a  quantity  consisting  of  nine  such  units 
and  one  unit  more. 

Counting  now  by  ten  units  at  a  time,  as  before  we  counted 

10 


NUMERATION  AND  NOTATION  11 

by  single  units,  we  get  the  numbers  ten,  twenty,  thirty,  forty, 
.  .  .,  ninety. 

The  names  of  the  numbers  between  ten  and  twenty  are,  in 
order :  eleven,  twelve,  thirteen,  fourteen,  .  .  .,  nineteen. 

The  names  of  the  numbers  between  twenty  and  thirty, 
thirty  and  forty,  .  .  .,  are  formed  by  placing  the  names  of 
the  numbers  one,  two,  three,  .  .  .,  nine,  in  order  after 
twenty,  thirty,  .  .  .,  ninety. 

14.  The  number  hundred  applied  to  any  unit  denotes  a 
quantity  which  consists  of  ten  ten-units. 

Counting  now  by  a  hundred  units  at  a  time,  as  before  we 
counted  by  single  units,  we  get  the  numbers  one  hundred, 
two  hundred,  .  .  .,  nine  hundred. 

The  names  of  the  numbers  between  one  hundred  and 
two  hundred,  two  hundred  and  three  hundred,  .  .  .,  are 
formed  by  placing  the  names  of  the  numbers  from  one  to 
ninety-nine  in  regular  succession  after  one  hundred,  two 
hundred,  .  .  .,  nine  hundred. 

15.  The  number  thousand  applied  to  any  unit  denotes  a 
quantity  which  consists  of  ten  hundred-units. 

Counting  now  by  a  thousand  units  at  a  time,  as  before  we 
counted  by  single  units,  we  get  the  numbers  one  thousand, 
two  thousand,  .  .  .,  nine  thousand,  ten  thousand,  eleven 
thousand,  twelve  thousand,  .  .  .,  twenty  thousand,  .  .  ., 
one  hundred  thousand,  .  .  .,  two  hundred  thousand,  .  .  ., 
nine  hundred  and  ninety-nine  thousand. 

The  names  of  the  numbers  between  one  thousand  and  two 
thousand,  two  thousand  and  three  thousand,  .  .  .,  are  formed 
by  placing  in  order  the  names  of  the  numbers  from  one  to 
nine  hundred  and  ninety-nine,  —  the  numbers  preceding  a 


12  ARITHMETIC 

thousand,  -  -  after    one    thousand,    two   thousand,  .  .  .,  nine 
hundred  and  ninety-nine  thousand. 

16.  The   number  million   applied  to   any  unit  denotes   a 
quantity  which  consists  of  a  thousand  thousand-units. 

The  number  billion  applied  to  any  unit  denotes  a  quantity 
which  consists  of  a  thousand  million-units. 

The  number  trillion  applied  to  any  unit  denotes  a  quantity 
which  consists  of  a  thousand  billion-units. 

17.  The  number  tenth  applied  to  any  unit  denotes  that 
quantity  of  which  ten  make  up  the  unit. 

The  number  hundredth  applied  to  any  unit  denotes  that 
quantity  of  which  ten  make  up  one  tenth  of  the  unit. 

Consequently,  one  hundred  of  the  hundredths  of  any  unit 
make  up  that  unit. 

The  number  thousandth  applied  to  any  unit  denotes  that 
quantity  of  which  ten  make  up  the  one  hundredth  of  the 
unit. 

Consequently,  one  thousand  of  the  thousandths  of  any  unit 
make  up  that  unit;  and  so  on. 

18.  We  count  by  the  tenth  of  a  unit  at  a  time,  as  before 
we   counted  from  one  to  nine  by  a  single  unit  each  time ; 
thus :  one  tenth,  two  tenths,  .  .  .,  nine  tenths. 

We  count  by  a  hundredth  of  a  unit  at  a  time,  as  before  we 
counted  from  one  to  ninety-nine  by  a  single  unit  each  time ; 
thus :  one  hundredth,  two  hundredths,  .  .  .,  ninety-nine 
hundredths. 

We  count  the  thousandths  of  a  unit,  from  one  to  nine 
hundred  and  ninety-nine  of  the  same  in  like  manner  as  we 
count  thousands  of  the  unit,  and  so  on. 


NUMERATION   AND   NOTATION  13 

19.  Notation  is  the  art  of  expressing  numbers  by  means  of 
certain  number  symbols  called  numerals  or  figures. 

20.  The  Arabic  Numerals,  styled  also  Figures,  are 

0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 
denoting  naught,  one,  two,  three,  four,five, six, seven, eight,nine 
respectively.  The  first  of  these  is  called  nauc/ht,  cipher,  or 
zero ;  the  remaining  nine  are  called  digits.  By  means  of 
these  numerals  and  a  dot  called  the  decimal  point,  we  can 
write  down  any  number  expressed  decimally.  The  method 
of  doing  so  may  be  described  as  follows : 

A  figure  immediately  to  the  left  of  the  decimal  point  de- 
notes so  many  single  units. 

A  figure  immediately  to  the  left  of  the  single-units  figure 
denotes  so  many  tens  of  the  units,  while  a  figure  immediately 
to  the  right  of  the  single-units  figure  denotes  so  many  tenths 
of  the  unit. 

Figures  to  the  left  of  the  tens-figure,  taking  them  in  order 
from  right  to  left,  denote  so  many  hundreds  of  the  unit,  so 
many  thousands  of  the  unit,  so  many  ten-thousands  of  the 
unit,  so  many  hundred-thousands  of  the  unit,  so  many  mil- 
lions of  the  unit,  etc. 

Figures  to  the  right  of  the  tenths-figure,  taking  them  in 
order  from  left  to  right,  denote  so  many  hundredths  of  the 
unit,  so  many  thousandths  of  the  unit,  so  many  ten-thousandths 
of  the  unit,  so  many  hundred-thousandths  of  the  unit,  so  many 
millionths  of  the  unit,  etc. 

The  function  of  the  decimal  point  is  to  mark  the  place  of 
the  standard  unit  when  the  quantity  is  measured. 

21.  Instead  of  speaking  of  "the  number  denoted  by"  5, 
75,  or  375,  we  may  for  brevity  speak  of  the  number  5,  75,  or 
375. 


14  ARITHMETIC 

22.  Consider  the  following  quantities : 

9  yd. 
59 
259 

3259  " 

43259  « 

843259  «                                            , 

59.7  "                                            1 

59.76  "                                            | 

59.761  «                                            I 

59.7613  «                                            ! 

59.76132  « 

9  denotes  9  of  the  unit  one  yd. 

5  "       5  "     "      "     ten  yd. 

2  "        2  "     "      "     one  hundred  yd. 

3  "        3  "     "      "     one  thousand  yd. 

4  "       4  "     "      "     ten  thousand  yd. 

8        "       8  "     "      "     one  hundred  thousand  yd. 

7  "        7  "     "      "     one  tenth  of  a  yd. 

6  "       6  "     "      "     one  one-hundredth  of  a  yd. 

1  "        1  "     "      "     one  one-thousandth  of  a  yd. 
3        "        3  "  "     one  ten-thousandth  of  a  yd. 

2  "        2  "     "      "     one  one-hundred  thousandth  of  a  yd. 

23.  The  number  8  always  denotes  8  of  the  unit ;  86  denotes 

8  of  the  ten-unit  or  80  of  the  unit  and  6  of  the  unit,  and  is 
read  eighty-six  of  the  unit. 

865  denotes  8  of  the  hundred-unit  and  6  of  the  few-unit  and 

5  of  the  unit ;  i.e.  800,  60,  and  5  of  the  unit,  and  is  read  eight 
hundred  and  sixty-five  of  the  unit. 

Thus,  the  numbers  8,  86,  865  always  denote  eight,  eighty- 


NUMERATION   AND   NOTATION  15 

six,  eight  hundred  and  sixty-five  respectively,  the  position  of 
the  figures  in  each  case  giving  the  unit.  For  example : 

In  the  number  865,865,865  each  865  is  read  eight  hundred 
and  sixty-five,  the  difference  being  in  the  unit  only ;  865  of 
the  million-unit,  865  of  the  thousand-unit,  and  865  of  the 
one-unit,  which  is  the  primary  unit  of  reference. 

This  number  is  read  eight  hundred  and  sixty-five  million 
eight  hundred  and  sixty-five  thousand  eight  hundred  and 
sixty-five. 

Exercise  3 

1.  Express  in  words  the  numbers  given  in  Exercises  11,  13, 
19,  20,  and  21. 

Express  in  words : 

2.  3,409,035;  42,590,709;  6,003,040. 

3.  20,394,678;  4,007,890;  8,000,006;  70,001,002. 

Exercise  4 

Express  in  words : 

1.  .7  ton;  .64  ton;  .643  ton.       3.    9.403  hr. ;  29.04  min. ;  .09  sec. 

2.  4.92  Ib. ;  8.09  Ib. ;  2.734  Ib.    4.    7.456  A. ;  6.7985  A. 

5.  8452.69  sq.  mi. ;  21.4394  A. 

6.  Express  in  words  the  numbers  given  in  Exercises  14  and  24. 

Exercise  5 
Write  in  figures : 

1.  Three  hundred  and  forty-nine  ;  eight  thousand  four  hundred 
and  sixty -nine ;  nine  thousand  five  hundred  and  seventy. 

2.  Twenty-nine  thousand  one  hundred  and  thirty-four;   fifty 
thousand  eight  hundred  and  seventy-six ;  seventy-eight  thousand 
three  hundred. 

3.  Nine  hundred  and  fifty-two  thousand  seven  hundred  and 
forty ;  six  hundred  and  forty-nine  thousand  nine  hundred  and 
five  ;  nine  hundred  thousand  eight  hundred  and  sixty-four. 


16  ARITHMETIC 

• 

4.  One  hundred  and  sixty-eight  thousand    six  hundred  and 
eighteen ;  three  hundred  and  twelve  thousand  seven  hundred  and 
forty-two;  four  hundred  and  sixty-one  thousand  eight  hundred 
and  twenty-one. 

5.  Seven  hundred  and  four  thousand  and  thirty;  three  hun- 
dred thousand  two  hundred  and  four ;  one  hundred  thousand  and 
fifty. 

6.  Five  million  two  hundred  and  ninety-five  thousand  three 
hundred  and  three. 

7.  Sixty -four  million  seven  hundred  and  eight  thousand  one 
hundred  and  thirty -four. 

8.  Seventy-eight  million  four  thousand  and  eighty-five. 

9.  Six  million  six  thousand  and  six. 

10.    What  is  the  value  of  six  thousand  units  of  §4  each  ?     Of 
six  million  units  of  $  5  each  ? 


Exercise  6 

Express  in  figures  : 

1.  Five  tenths;  two,  and  sixty-seven  hundredths;  nine  hun- 
dredths ;  four  thousandths. 

2.  Nine  hundred  and  twelve  thousandths  ;  seven  hundred  and 
four  thousandths;    five  thousand  four  hundred  and  sixteen  ten 
thousandths. 

3.  Four  hundred  and  fifty-three,  and  four  hundredths;  four 
hundred  and  fifty,  and  one  hundred  and  twenty-six  thousandths. 

4.  Nine  hundred  and  six  thousandths ;  nine  hundred  and  six 
ten  thousandths;  twenty,  and  forty -five  thousandths. 

5.  Seventeen,  and  seven  ten  thousandths;  three  hundred  and 
three,  and  nine  hundred  and  nine  ten  thousandths. 


NUMERATION   AND   NOTATION  17 

• 

Exercise  7 

1.  Name  each  unit  in  346  yd.  and  state  the  number  of  units  of 
each  kind. 

2.  In  question  1,  what  is  the  ratio  of  each  unit  to  the  next 
unit  to  the  right  of  it  ? 

3.  What  part  is  each  unit  of  the  next  unit  to  the  left? 

4.  What  quantity  expressed  in  single  units  is  indicated  by  3 
in  question  1  ?     By  4  ?     By  6  ? 

5.  Express  in  words  3.46  feet.     Name  each  unit. 

6.  Fill  in  the  blanks:    $  679  is  equal  to  6  units  of         -, 
7  units  of ,  and  9  units  of  -  — . 

7.  Write  as  one  number,  8  of  the  hundred-unit,  5  of  the  ten- 
unit,  and  6  of  the  one-unit. 

8.  Name  the  two  chief  units  in  843,294  and  state  the  number 
of  units  of  each  kind.     Express  the  number  in  words. 

9.  Write  the  name  of  each  of  the  six  units  in  $294,785  and 
the  number  of  each  unit. 

10.  In  the  quantity  $666,666,  which  6  represents  the  largest 
sum  of  money  ?     Which  the  smallest  ? 

1 1 .  Write  as  one  number  67  of  the  thousand-unit  and  413  of 
the  one-unit. 

12.  Write  as  one  number  5  of  the  million-unit,  463  of  the 
thousand-unit,  and  768  of  the  one-unit. 

13.  Write  as  one  number  349  of  the  thousand-unit  and  258  of 
the  one-unit. 

14.  Write  as  one  number  365  of  the  million-unit,  829  of  the 
thousand-unit,  and  604  of  the  one-unit. 

15.  Which  is  the  largest  unit  used  in  giving  the  population  of 
the  largest  city  in  the  United  States  ?     The  length  of  the  largest 
river?     The  height- of  the  highest  mountain?     The  area  of  the 
largest  state  ?     The  area  of  the  smallest  state  ? 

c 


18  ARITHMETIC 

THE  ROMAN  NOTATION 

24.  The  Arabic  Notation  is  the  one  in  general  use.     It 
was  introduced  into  Europe  by  the  Arabs.     The  system  of 
notation  which  was  used  among  the  Romans  is  now  used  only 
to  denote  the  chapters  and  sections  of  books,  etc. 

25.  The  following  letters  are  used  to  denote  numbers,  and 
their  values  are  written  below: 

I.        V.        X.        L.         C.        D.        M. 

1.         5.        10.        50.       100.     500.     1000. 

26.  The  numbers  6,  8,  15,  20  are  represented  thus  : 

VI.       VIII.       XV.       XX. 

Hence  if  a  character  in  the  Roman  Notation  be  followed  by 
another  of  equal  or  less  value,  the  number  denoted  by  the  ex- 
pression is  equal  to  the  sum  of  the  simple  values. 

27.  The  numbers  4,  9,  40,  and  90  are  represented  by  IV., 


Hence  if  a  character  in  the  Roman  Notation  is  followed  by 
one  of  greater  value  than  itself,  the  number  denoted  by  the 
expression  is  the  difference  of  their  simple  values. 

28.   Express  1896  in  Roman  numerals. 

1896  =  1000,  800,  90,  and  6. 
1000  =  M. 
800  =  DCCC. 
90  =  XC. 
6  =  VI. 
,-.  1896  -  MDCCCXCVI. 


NUMERATION   AND  NOTATION  19 

Hence  to  write  any  number  in  Roman  numerals,  separate 
the  number  into  its  different  parts,  and  write  down  the  parts 
in  order,  beginning  at  the  left. 

Exercise  8 

Write  in  Roman  numerals : 

1.  14,  25,  54,  89,  99. 

2.  178,  304,  871,  982,  999. 

3.  1204,  1590,  1756,  1876,  1895. 

Write  in  figures : 

4.  XL VI.,  LXXIX.,  XCIV.,  LXXXIII. 

5.  XCIX.,  CXXXIX.,  CLX. 

6.  DLIV.,  MDCIL,  MDCCCXIX.,  MXC. 


CHAPTER   III 
ADDITION 

29.  Let  the  length  of  a  room  be  measured  by  the  parts, 
2  ft.,  3  ft.,  4  ft.,  and  5  ft.     Here  the  common  unit  of  meas- 
ure, 1  ft.,  has  been  repeated  2,  3,  4,  and  5  times  to  measure 
the  parts. 

The  number  of  units  in  all  is  the  sum  found  primarily  by 
counting  2,  3,  4,  and  5,  or  14  units  of  1  ft.  Hence  the 
length  of  the  room  which  is  now  definitely  measured  is 
14  ft. 

Addition  may,  therefore,  be  considered  as  the  operation  of 
finding  the  quantity,  which,  as  a  whole,  is  made  up  of  two  or 
more  given  quantities  as  its  parts.  Each  of  these  quantities 
must  have  the  same  measuring  unit.  Not  only  is  it  im- 
possible to  add  5  ft.  to  4  min.,  it  is  impossible  to  add  5  ft. 
to  4  in. ;  i.e.  to  express  without  change  of  unit  the  whole 
quantity  by  a  number  of  either  ft.  or  in. 

The  parts  added  are  called  Addends. 

The  Sum  is  the  quantity  obtained  by  adding  the  quantities 
expressed  in  terms  of  a  common  unit. 

30.  The  Sign  of   Addition  is  +,  and  is  read  plus;   thus 
6  -|-  8  is  read  6  plus  8. 

The  Sign  of  Equality  is  =,  and  is  read  equals  or  equal; 
thus  4  +  5  =  9  is  read  4  plus  5  equals  9. 

31.  I  bought  3  farms  of   50  A.  each,  6  farms  of  50  A., 
and  4  farms  of  50  A.     How  much  did  I  buy  altogether  ? 

20 


ADDITION  21 

Here  we  are  required  to  find  the  whole  quantity  measured  by  the  sum  of 
3,  6,  and  4  farms  of  50  A. 

.•.  the  whole  quantity  =  13  farms  of  50  A. 

Exercise  9 

1.  What  quantity  is  measured  by  the  parts  2  yd.,  6  yd., 

and  7  yd.  ? 

2.  What  sum  of  money  is  equal  to  4  five-cent  pieces,  9  five- 
cent  pieces,  and  5  five-cent  pieces  ? 

3.  How  much  is  8  fifty -dollar  bills,  4  fifty-dollar  bills,  and 
9  fifty-dollar  bills  ? 

4.  I  paid  out  in  one  day  6  ten-dollar  bills,  8  ten-dollar  bills, 
and  5  ten-dollar  bills.     How  much  did  I  spend  altogether  ? 

5.  If  I  sell  two  lots,  one  for  8  units  of  value,  and  the  other 
for  6  units,  what  do  I  get  for  both,  the  unit  of  value  being  $  100  ? 

6.  A  fruit  dealer  who  arranges  his  apples  in  piles  of  6  for  5 
ct.  sells  1  pile  to  each  of  a  company  of  4  persons,  and  3  piles  to 
another  customer.     How  much  does  he  sell  altogether? 

7.  A  speculator  buys  5  farms  of  100  A.  for  $5000,  6  farms 
of  100  A.  for  $  7000,  and  3  farms  of  100  A.  for  $  4000.    How 
much  land  did  he  buy  ?    If  $  1000  is  the  unit  of  value,  how  many 
units  did  he  pay  out  for  all  the  farms  ? 

8.  What  is  the  quantity  denoted  by  the  sum  6,  7,  and  5  times 
the  measuring  unit  ? 

9.  2  in.  +  5  in.  +  4  in.  =  ?     2  ft.  +  5  ft.  +  4  ft.  =  ? 

10.  $3  +  $4  +  $6  =  ?     3  ten-dollar  bills  +  4  ten-dollar  bills 
+  6  ten-dollar  bills  =  ? 

11.  A  horse  was  bought  for  10  ten-dollar  bills,  and  a  carriage 
for  12  ten-dollar  bills.     How  much  was  paid  for  both  ? 

12.  A  man  paid  out  at  one  time  4  five-dollar  bills,  at  another 
6  five-dollar  bills,  and  again  3  five-dollar  bills.     How  much  did 
he  pay  out  altogether  ? 


22  ARITHMETIC 

13.  A  fruit  dealer  arranges  his  fruit  in  piles  of  3  each.     He 
has  20  piles  of  apples,  10  of  oranges,  and  30  of  plums.     How 
much  has  he  altogether  ?     If  there  were  30  of  each  kind  in  a  pile, 
what  number  would  express  the  aggregate  ? 

14.  A  fruit  dealer  sells  his  apples  at  the  rate  of  3  for  5  cents. 
He  sells  five  cents'  worth  to  each  of  8  customers.    How  much  did 
he  sell  ? 

15.  What  is  the  sum  of  two  quantities,  one  denoted  by  9  times 
the  measuring  unit,  and  the  other  by  6  times  the  measuring  unit? 

16.  If  I  paid  30  units  of  $  100  each  for  a  lot,  and  built  a  house 
upon  it  which  cost  me  40  units  of  $  100  each,  what  was  the  total 
cost  ?     If  $  1000  is  the  unit,  what  is  the  number  expressing  the 
total  cost  ? 

17.  A  horse  which  travels  at  the  rate  of  8  mi.  an  hr.  goes 
from  A  to  B  in  3  hr.,  from  B  to  C  in  2  hr.,  and  from  C  to  D 
in  4  hr.      If  8  mi.  is  the  unit  of  length,  what  is  the  number 
expressing  the  distance  from  A  to  D  ? 

32.    Drill  on  the  following  addition  combinations  to  secure 
accuracy  and  rapidity : 

1112      12       123       123       1234 

1     2    3     2.     4    3.     5    4    3.     6     5    4.     7     6     5     4. 

_,_,_,_,     -,    - ,     _,_,_,     _,_,_,     _,_,_,_, 

1234   12345   2345   3456 

8765.  98765.  9876.  9876. 

-J       — >       -)       —  )         — >        — >         ->        —  1        —  1         —  >       —5        -J       —  >          —)       —)       —)       —  » 

456      567       67       78      8       9 

9    8     7.     9    8     7.     9     8.     9    8.     9.     9 

_,_,_,     _,_,_,     _,    _ ,     __,    _ ,     _ ,     _. 

Enlarge  each  combination  thus : 

888  8     18     28  18     38     68 

?    1?    ?9,  etc.;    2    J     4  etc. ;     1?    29     39    etc. . 
80      80  80     180     280 

etc.-     90     J!5      9Q;  etc. 


ADDITION  23 

Give  such  problems  as  the  following  requiring  instantane- 
ous answers : 

How  many  sq.  ft.  in  1  sq.  yd.  8  sq.  ft.?  1  sq.  yd.  6  sq. 
ft.?  etc. 

How  many  qt.  in  1  pk.  4  qt.  ?  etc. 

When  7  is  the  number  added,  base  the  problem  on  days ; 
thus: 

How  many  da.  in  1  wk.  4  da.  ? 

When  the  number  is  6,  on  minutes  ;   thus : 

How  many  sec.  in  1  min.  30  sec.,  and  so  on. 

Exercise  10 

Using  any  unit,  count  to  the  number  next  larger  than  100. 
By: 

1.  2'sfromO;  from  1. 

2.  3's  from  0;  from  2. 

3.  4's  from  0 ;  from  1,  2,  and  3  separately. 

4.  5's  from  0 ;  from  1,  2,  3,  and  4. 

5.  6's  from  0,  1,  2,  3,  4,  and  5. 

6.  7's  from  0,  1,  2,  3,  4,  5,  and  6. 

7.  S's  from  0,  1,  2,  3,  4,  5,  6,  and  7. 

8.  9's  from  0,  1,  2,  3,  4,  5,  6,  7,  8,  and  9. 

33.  A  person  paid  $38  for  a  cow,  $146  for  a  horse,  and 
$  255  for  a  carriage.  Find  the  cost  of  all. 

$    38  ^n  this  problem  we  are  required  to  find  the  cost  which  is  the 

1  4fi       "whole  measured  by  the  parts  $  38,  $  146,  and  $  255. 

This  may,  for  convenience,  be  broken  up  into  the  sum  of  5,  6, 
and  8  units  of  $1,  5,  4,  and  3  units  of  $10,  and  2  and  1  units 


$  439       of  $  100'     The  sum  of  5'  6'  and  8  units  of  ® 1  =  10  units  of  ®  1  = 
unit  of  $  10  and  9  of  the  $  1  unit. 

Add  the  1  unit  of  $  10  in  with  the  tens'  column. 


24  ARITHMETIC 

The  sum  of  1,  5,  4,  and  3  units  of  $  10  =  13  units  of  $  10,  or  1  unit  of 
$100,  and  3  units  of  $10. 

Add  the  1  unit  of  $  100  in  with  the  hundreds'  column. 

The  sum  of  1,  2,  and  1  unit  of  $  100  =  4  units  of  $  100. 

Hence  the  cost  =  4  units  of  $  100,  3  units  of  $  10,  and  9  units  of  $  1  =  $  439. 

34.  If  the  primary  unit  is  1  da.,  and  the  derived  unit 
1  yr.,  how  many  primary  units  of  time  are  there  in  one 
derived  and  243  primary  units? 

Here  we  are  required  to  find  the  sum  of  365  and  243  primary  units.  The 
result  —  608  primary  units. 

Find  the  sum  of  the  following  numbers,  using  the  unit 
employed  in  stating  your  age : 

234  •'•  tne  sum  =  2626  yr.,  since  1  yr.  is  the  unit  of  age. 

Add  thus,  beginning  at  the  units'  column :  9,  12,  16  ;  write  down 
6  and  carry  1  to  the  tens'  column;  5,  11,  19,  22;  write  down  2 
under  the  tens'  column  and  carry  2  to  the  hundreds'  column;  10, 15, 
24,  26  ;  write  down  26,  putting  the  6  under  the  hundreds'  column. 
To  prove  the  answer  correct,  add  downward.  If  the  same 
answer  is  obtained,  the  result  is  likely  to  be  correct. 

Exercise  11 

Find  the  sum  of  the  following  quantities  and  explain  your 
work  clearly.  Name  all  the  primary  units  and  also  the  quanti- 
ties which  measure  the  parts.  Prove  each  answer  correct  by 
beginning  at  the  top  and  adding  down. 

1.    $441  2.    341  ct.  3.    532  da. 

234  225  "  233   " 

518  343  "  154   « 


4.       543  min.  5.    635  hr.  6.    247  in. 

666  "  87  "  859  " 

752  «  256  "  23  « 

231  "  742  "  271  " 


ADDITION 


25 


7.    2576ft. 
3491  « 
7743  « 

8988  « 


8.    2598yd. 
6776    " 
4259    « 
7362    " 


9.    4397  mi. 
8999    « 
5637    « 
8249    « 


10.    64251  oz. 

3789   « 

45278   " 

99   « 

6472   « 


11.    89435  Ib. 

62789  " 

576  " 

43  " 


13.  $453798 
667788 
549763 

438925 
648888 
999999 


12.  850439  T. 
973642  " 
845867  « 
939894  « 
768795  « 
649879  « 


14.  Find  the  number  of  days  in  1  leap  yr.  and  257  da. 

15.  If  the   unit   of   length   1   ft.   is  repeated  5280   times  to 
measure  a  mi.,  how  many  such   units   are  there   in   1  mi.  and 
4754  ft.  ?     In  1  mi.  1  yd.  2  ft.  ? 

16.  If  1  mi.   is  represented  by  the  number  63,360  when  the 
primary  unit  is  1   in.,   how  many  times  will  the   primary  unit 
measure  the  distance  1  mi.  5769  in.  ?      The  distance  1  mi.  1  yd. 
1  ft.  8  in.  ? 

Find  the  sum  of  the  following  quantities,  the  unit  of  measure 
being  that  in  selling  retail  the  following :  In  (17)  cherries, 
(18)  kerosene,  (19)  raspberries,  (20)  sugar,  (21)  coal,  (22)  cloth, 
(23)  potatoes,  (24)  eggs. 


26 


ARITHMETIC 


17.  84 

18.  99 

19.  893 

20.  733 

93 

77 

254 

842 

89 

86 

767 

951 

75 

25 

899 

258 

91 

43 

654 

365 

27 

88 

473 

874 

30 

76 

129 

935 

45 

52 

895 

273 

21.  542 

22.  9834 

23.  7594 

24.  5846 

879 

729 

821 

7593 

666 

8345 

2357 

3819 

257 

728 

8463 

5578 

389 

3403 

1525 

2904 

983 

17 

7469 

8392 

365 

295 

2856 

9576 

874 

8943 

8888 

2882 

35.    If  the  unit  1  mi.  contains  5280  ft.,  how  many  ft.  are 
there  in  4  such  units  ? 

Here  we  are  required  to  find  the  sum  of  four  equal  addends,  each  of  which 

is  5280  ft. ;  thus  : 

5280  ft. 

5280  " 

5280  " 

5280  " 

21120  ft. 

Exercise  12 

1.  If  1  mi.  contains  1760  yd.,  find,  by  adding,  the  number  of 
yd.  in  2  mi.     In  3  mi.     In  4  mi. 

2.  If  the  unit  1  mi.  contains  320  rd.,  how  many  rd.  in  9  such 
units  ? 

3.  One  sq.  ft.   contains  144  sq.   in.      How  many  sq.  in.    are 
there  in  two  oblongs,  one  containing  3   sq.    ft.    and   the   other 
4  sq.  ft.? 


ADDITION  27 

4.  One  cu.  ft.  contains  1728  cu.  in.     Show  by  adding  that  12 
volumes,  each  containing  144  cu.  in.,  will  be  equal  to  1  cu.  ft. 

5.  One  gal.  contains  231  cu.  in.     Show  that  a  gallon  measure 
can  be  filled  with  water  and  emptied  7  times  into  a  measure 
containing  1  cu.  ft.  without  causing  it  to   overflow,  but  not  8 
times. 

6.  If  the  unit  of  area,  1  A.,  contains  4840  sq.  yd.,  how  many 
sq.  yd.  are  there  in  6  such  units  ? 

7.  If  there  are  197  school  days  in  1  yr.,  find,  by  adding,  the 
number  of  school  days  in  8  yr. 

8.  One  sq.  mi.  contains  640  A.     How  many  A.  are  there  in 
8  sq.  mi.  ? 

9.  Find,  by  adding,  the  number  of   da.  in  6  ordinary  yr.  of 
365  da.  each  and  2  leap  yr.  of  366  da.  each. 

Exercise  13 

In  each  of  the  following  questions,  state  in  each  case  which  is 
the  whole  quantity  to  be  measured  and  what  are  the  parts  meas- 
uring it. 

1.  A  spent  the  following  sums  of  money:  $425,  $342,  $673, 
and  $  897.     How  much  did  he  spend  altogether  ? 

2.  An  encyclopaedia  consists  of  three  volumes.     In  the  first 
there  are  693  pages,  in  the  second  745,  and  the  third  892.     Find 
the  number  of  pages  in  the  encyclopaedia. 

3.  Using   the   table   in    §  171,    find   the   number    of   days    in 
the  first  six  months  of  the  year.     In  the  last  six  months.     In  a 
year. 

4.  Find  the  number  of  days  in  the  three  spring  months.     In 
the  three  summer  months.      In  the  three  fall  months.     In  the 
three  winter  months, 


28  ARITHMETIC 

5.  If  the  average  number  of  persons  to  the  sq.  mi.  is  taken 
as  the  unit  of  population,  and  the  unit  for  California  is  8  persons, 
what  is  the  unit  of  population  for  the  following  states  ? 

A,  which  has  twice,  and  B,  which  has  three  times  the  average 
population  of  California ;  C,  whose  unit  is  the  sum  of  the  units  of 
A  and  B. 

6.  Find  the  total  area  of  these  lakes : 

Lake  Erie,  area    7,750  sq.  mi. ; 

Lake  Ontario,  area  6,950  sq.  mi. ; 
Lake  Michigan,  area  22,000  sq.  mi. ; 
Lake  Superior,  area  31,500  sq.  mi. 

7.  A  merchant  bought  150  yd.  of  cloth  for  $  232,  254  yd.  for 
$  175,  1875  yd.   for  $  2395,  and  640  yd.  for  $  1966.     Find  the 
number  of  yd.  bought  and  the  total  cost. 

8.  How  many  years  are  there  between  the  establishment  of 
the  Republic  of  Rome  in  509  B.C.,  and  the  Declaration  of  Inde- 
pendence in  1776  A.D.  ? 

9.  What  is  the  sum  of  4,  6,  9,  7,  5,  8,  3,  6,  and  7,  when  the 
unit  of  value  is  $  1  ?  $  10  ?  $100? 

10.  What  is  the  value  of  the  quantity  denoted  by  the  sum  of  3, 
9, 12,  6,  and  the  unit  $  1000  ?  $  10,000  ?  $  100,000  ?  $  1,000,000  ? 

11.  Find  the  sum  of  three  hundred  and  seventy-six  thousand 
and  fifty-four ;  one  hundred  and  ninety-seven  thousand  two  hun- 
dred and  fifty-one ;    four  hundred  and  fifty-seven   thousand   six 
hundred  and  forty-nine. 

12.  The  population  of  Maine  is  661,086;    New  Hampshire, 
376,530;    Vermont,   332,422;    Massachusetts,   2,238,943;    Rhode 
Island,  345,506,  and  Connecticut,  746,258.     Find  the  population 
of  the  New  England  States. 

13.  According  to  the  census  of  1890,  the   population  of  the 
seven  largest  cities  in  the  United  States  is  :  New  York,  1,515,301 ; 
Chicago,  1,099,850;  Philadelphia,  1,046,964;  Brooklyn,  806,343; 


ADDITION  29 

St.    Louis,   451,770;    Boston,   448,477,   and   Baltimore,   434,439. 
Find  the  total  population. 

14.  Find  the  sum  of  the  following  measures:  one  1-inch  unit, 
one  12-inch  unit,  one  36-inch  unit,   one  198-inch  unit,  and  one 
63360-inch  unit. 

15.  Find  the  sum  of  the  following:  2  of  the  one-unit,  4  of  the 
ten-unit,  6  of  the  hundred-unit,  3  of  the  thousand-unit,  and  9  of 
the  ten-thousand  unit  quantities. 

16.  Find  the  number  of  times  a  clock  strikes  from  a  quarter  of 
nine  A.M.  until  a  quarter  of  nine  P.M. 

17.  A,  B,  and  C  engaged  in  trade  ;  A  put  in  $  3475,  B  $  4593, 
and  C  as  much  as  the  other  two  together.     How  much  money  was 
put  into  the  business  ? 

18.  A  man,  dying,  willed  to  his  widow  $  6875 ;   to  his  son, 
$  4294,  and  to  his  daughter,  $  3875.     What  was  the  value  of  his 
estate  ? 

19.  The  census  of  1890  gave  the  negro  population  of  Georgia 
858,996;    Mississippi,  744,749;    South   Carolina,  689,141;    Ala- 
bama,  679,299;     Virginia,   635,858;     North    Carolina,   562,565; 
Louisiana,  560,192 ;  Texas,  489,588 ;  Tennessee,  430,881 ;  Arkan- 
sas, 309,427.     Find  the  entire  negro  population  of  these  states. 

20.  The  area  of  the  basin  of  the  Colorado  River  is  250,000 
sq.   mi. ;    Columbia,   250,000 ;    Mackenzie   River,   440,000 ;   Mis- 
souri-Mississippi,   1,250,000 ;     Nelson,    355,000 ;      Rio    Grande, 
180,000;    St.  Lawrence,  350,000.     Find  the  total  area  of  these 
river  basins. 

21.  What  is  the  area  of  the  New  England  States;    that  of 
Maine   being   in   sq.   mi.    33,040,  of    New    Hampshire   9305,    of 
Vermont  9565,  of  Massachusetts  8315,  of  Rhode  Island  1256, 
of  Connecticut  4990  ? 

22.  A  father  left  his  eldest  son  $  24,000  more  than  he  left  his 
second  son,  and  the  second  son  $  7560  more  than  the  third ;  to 
the  third  he  left  $  60,480.     What  was  the  eldest  son's  portion, 
and  what  sum  did  the  father  leave  to  his  three  sons  ? 


30 


ARITHMETIC 


23.  If  in  question  6,  100  sq.  mi.  =  unit  of  measure,  and  in 
question  20,  1000  sq.  mi.   =  the  unit,  what  numbers  express  the 
respective  aggregates  ? 

24.  Add  vertically  and  horizontally  the  following  statement  of 
eight  weeks'  cash  receipts  : 


MON. 

TUES. 

WED. 

THUR. 

FBI. 

SAT. 

TOTAL. 

1st 

$3862.93 

$1391.76 

$6760.68 

$1098.91 

$1696.65 

$     43.68 

2d 

396.74 

6168.37 

864.39 

964.26 

167.69 

1864.86 

3d 

1768.63 

467.89 

2035.68 

3165.03 

691.83 

785.97 

4th 

3976.98 

76.05 

364.76 

93.68 

1948.39 

1759.46 

5th 

263.76 

1035.84 

36.10 

386.41 

3.45 

1396.71 

6th 

1559.83 

1932.57 

1268.15 

8.37 

279.72 

67.85 

7th 

62.24 

318.62 

134.36 

1763.29 

1468.29 

543.66 

8th 

194.87 

3.85 

7643.82 

685.38 

765.42 

39.67 

Total 

25. 

Add  vertically 

and 

horizontally 

the 

following 

statement  : 

TOTAL. 

$1169.84 

$3650.12 

$  189.10 

$  97.22 

$  26.55 

$  851.02 

909.58 

866.78 

914.19 

239.49 

297.02 

312.60 

575.72 

742.49 

1654.70 

196.17 

859.69 

1477.42 

2678.28 

1180.66 

119.25 

8418.60 

2223.42 

568.35 

312.83 

1638.24 

2016.72 

1542.24 

5300.20 

116.02 

1052.47 

342.65 

108.00 

349.95 

136.97 

1214.03 

339.11 

687.23 

215.17 

1020.00 

1124.50 

1732.25 

1732.50 

514.02 

557.60 

600.00 

475.00 

138.50 

1237.50 

3839.25 

777.60 

136.70 

4656.65 

1097.47 

113.56 

1291.98 

112.50 

1850.14 

738.75 

1204.74 

3661.00 

973.03 

311.20 

636.99 

243.44 

142.91 

1139.67 

670.22 

1201.64 

7357.51 

252.47 

694.62 

Total 

ADDITION  31 

36.   Find  the  sum  of:   2.46,  23.973,  15.025,  643.319,  and 

.468. 

2.46 

QQ  qyo  Since  we  can  add  numbers  of  the  same  unit,  we  write  the 

addends  so  that  units  will  be  under  units,  tenths  under  tenths, 

and  so  on.     This  is  easily  done  by  placing  the  decimal  points 

643.319        directly  below  each  other.     Then,  beginning  at  the  right,  we 

.468        ac^  tne  n»ures  as  if  they  were  integers,  and  place  the  decimal 

point  in  the  sum  between  the  units'  and  tenths'  column. 
68O.245 

Exercise  14 

Add: 

1.    3.456  2.    27.43  3.    76.425 

4.593  18.314  39.639 

7.245  5.687  28.764 

9.864  34.986  21.385 


Write  in  columns  and  add : 

4.  4.396  +  7.295  +  6.478  +  5.766. 

5.  .432  +  .987  +  .593  +  .666. 

6.  84.63  +  46.892  +  24.7  +  95.657. 

7.  $  24.375 +  $  95.875  +  f  16.125 +  $  19.50. 

8.  Find  the  capacity  of  four  bins,  the  first  of  which  will  con- 
tain 66.384  bu.,  the  second  89.645  bu.,  the  third  27.437  bu.,  and 
the  fourth  75.938  bu. 

9.  What  is  the  area  of  a  farm  which  is  divided  into  three 
fields  containing,  respectively,  25.936  A.,  14.56  A.,  and  24.504  A.  ? 

10.  Four  towns,  A,  B,  C,  D,  lie  on  a  road  running  directly 
east  and  west.  The  distance  from  A  to  B  is  5.693  mi.,  from  B 
to  C  8.421  mi.,  from  C  to  D  12.768  mi.  Find  the  distance  from 
A  to  D, 


CHAPTER   IV 

SUBTKACTION 

37.  A  man  who  earned  $14  a  week,  spends  $  5  a  week  for 
his  board.     How  much  has  he  left  ? 

We  are  here  given  the  whole  quantity,  or  $14,  and  one 
part,  or  $  5,  and  we  are  required  to  find  the  other  part. 

The  question  may  be  viewed  in  two  ways:  How  much 
must  be  added  to  1 5  to  make  $  14  ?  Or  how  much  must  be 
taken  from  $  14  to  leave  $  5  ?  The  answer  in  both  cases  is 
known  from  addition.  $5  and  $9  are  two  quantities  making 
$  14.  Therefore,  if  one  of  them,  $  5,  is  given,  the  other  must 
be  $9.  Or,  in  other  words,  $9  is  the  difference  between 
$14  and  $5.  It  is  this  view  of  difference  that  gives  the 
name  Subtraction. 

38.  Subtraction  may  therefore  be  defined  as  the  operation 
of  finding  the  part  of  a  given  quantity  that  remains  when  a 
given  part  has  been  taken  from  the  quantity. 

The  given  quantity  is  called  the  Minuend,  and  the  given 
part  the  Subtrahend,  while  the  part  that  remains  is  called 
the  Difference  or  Remainder. 

39.  The  Sign  of  Subtraction,  — ,  is  called  minus.     Thus 
8  -  -  6  is  read  8  minus  6,  and  signifies  that  6  is  to  be  sub- 
tracted from  8. 

32 


SUBTRACTION  33 

Exercise  15 

Read  the  following  questions,  filling  in  the  blanks  : 

1 .  6  and  7  are  — ,       8  and  9  are  -  — ,  4  and  8  are  — . 

2.  4  and  6  are  — ,       4  and  —  are  10,  9  and  —  are  15. 

3.  2  and  —  are  11,     3  and  —  are  8,  6  and  —  are  14. 

4.  22  and  —  are  25,   4  and  -  -  are  36,  9  and  -  -  are  27. 

5.  8  and  —  are  29,     5  and  —  are  16,  6  and  —  are  48. 

Subtract  (Note :  Let  the  process  be  not  6  from  9  leaves  3,  but 
6  and  3  are  9) : 

9.    19.    29.    39.    49.    69.    99 
6'      6'      6'      6'      6'      6'      6* 

90.  190.    8.    28.    80.    380 

'60'  60'    5'      5'    50'      50* 

12.  120.    32.    320.    14.    54 

7'  70'      7'      70'      8'      8* 

What  numbers  added  respectively  to  9,  7,  6,  8,  5,  and  4,  make 
9.    12?       10.    15?       11.   17?       12.    14?       13.   18?       14.   16? 

40.  Drill,  as  in  §  32,  in  addition,  on  the  fundamental  sub- 
tractions,   connecting   with    corresponding    additions,    until 
accuracy  and  rapidity  are  secured ;  thus  : 

9     19     29     39     49     59 

-;       -;       -;       -;       -;       - :  and  so  on. 

8'      8'      8'      8'      8'      8' 

90     190     290     390     490 

-  :         -  ;         -  ;        —  ;         -  ;  and  so  on. 

80       80       80'      80'      80' 

17      27      37      47      57 

-  :     — :     — :       -  ;       -  ;  and  so  on. 

8'      8'      8'      8'      8' 

41.  A  man  who  owned  18  farms  of  50  A.,  sold  7  of  them. 
How  much  had  he  left  ? 

Because  7  +  11  =  18,  it  is  evident  that  he  had  left  11  farms  of  50  A. 
each. 

D 


34  ARITHMETIC 

Exercise  16 

1.  9  ft.  +  ?  =  16  ft.     9  yd.  +  ?  =  16  yd. 

2.  How  many  dimes  must  be  added   to  6  dimes  to  get  14 
dimes  ?     What  is  the  difference  between  14  dimes  and  6  dimes  ? 
How  much  less  is  6  five-dollar  bills  than  14  five-dollar  bills  ? 

3.  A  fruit  dealer  sold  10  piles  of  3  apples  each.    How  many 
had  he  left  if  he  had  at  first  15  piles  of  3  apples  ? 

4.  A  fruit  dealer  arranges  his  oranges   into  12  piles  of   4 
oranges  each.     He  sells  8  piles.     How  many  has  he  left  ? 

5.  A  fruit  dealer  who  sells  apples  at  the   rate  of   3  for   5 
ct.,  arranges   his   apples   into   19   groups  of  3   each.     He   sells 
5  ct.  worth  to  each  of  12  customers.     How  much  has   he   still 
remaining  ? 

6.  I  owe  a  debt  of  12  ten-dollar  bills  and  have  5  ten-dollar 
bills  in  my  pocket.     If  I  pay  this  toward  the  debt,  how  much 
do  I  still  owe  ? 

7.  What  must  be  added  to  15  units  to  get  20  units?     Taken 
from  20  units  to  get  15  units  ?     To  get  5  units  ?     If  I  sell  my 
house,  which  cost  20  units  of  value  of  $  100  each,  for  25  units, 
how  much  did  I  gain  on  the  transaction  ? 

8.  What  is  the  difference  between  a  quantity  denoted  by  14 
times  the  measuring  unit  and  one  denoted  by  8  times  the  meas- 
uring unit  ? 

9 .  A  person  who  has  $  50  pays  a  debt  of  $  30.     How  much 
money  has  he  left  ?     What  number  expresses  this  remainder  if 
$  10  is  the  unit  of  measure  ?      If  $  5  is  the  unit  of   measure  ? 
If  $  4  is  the  unit  ?     If  $  2  is  the  unit  ? 

10.  A  speculator  bought  12  farms  of  100  A.  each  for  $6000. 
He  sold  4  of  these  farms  for  $  3000.  How  much  land  had  he 
left  ?  If  $  1000  is  the  unit  of  money,  express  in  terms  of  this 
unit  the  difference  between  the  buying  and  the  selling  price  of 
the  4  farms. 


SUBTRACTION  35 

42.  The  following  method  of  subtraction,  which  is  nearly 
always   adopted   in   making    change,   is   almost   universally 
employed  by  professional  computers  and  is  considered  by 
many  teachers  the  best  way  to  perform  subtraction. 

It  is  superior  in  accuracy  and  rapidity  to  the  method  of 
the  next  paragraph. 

It  is  based  on  the  principle  that  the  sum  of  the  subtrahend 
and  remainder  is  equal  to  the  minuend. 

From  875  take  451.  Thus :  1  and  4  are  5  ;  5  and  2  are  7  ;  4  and 

_£  4  are  8.     In  this  operation  let  the  pupil  fancy 

that  he  is  doing  addition  with  the  sura  at  the 

451  top,  and  as  he  works  set  down  the  figures,  4,  2, 

424  and  4. 

43.  A  merchant  bought  965  yd.  of  silk  and  sold  723  yd. 
How  much  had  he  left  ? 

Here  we  are  required  to  find  the  undefined  part.  This  is  the  difference 
between  the  measured  whole,  or  965  yd. ,  and  the  given  part,  723  yd. 

965  yd.  =  the  measured  whole. 

723  yd.  =  the  measured  part. 

242  yd.  =  the  difference,  which  is  now  definitely  known. 

EXPLANATION. — As  in  addition,  we  write  units  under  units,  tens  under 
tens,  and  hundreds  under  hundreds.  Beginning  with  the  units,  we  say  3  units 
from  5  units  leaves  2  units,  which  we  write  below  the  line  in  the  units'  column. 
Then  2  tens  from  6  tens  leaves  4  tens.  Place  this  in  the  tens'  column. 
Lastly,  7  hundreds  from  9  hundreds  leaves  2  hundreds,  which  we  write  in  the 
hundreds'  column. 

This  difference,  242  yd.,  is  the  other  part,  which  is  now  definitely 
measured. 

Exercise  17 

t 

Subtract,  and  prove  your  answer  correct  in  each  case : 

j     $946  785  Ib.  659  T  897  da. 

324  '    323    "  '   236  "  '    683    " 


36  ARITHMETIC 

8498  hr.          9999  min.  8395  sec. 

'  2361  "         '  7265   "          '  4073  « 

$7684  $8697  $2578 

6450  1082  1506 

Exercise  18 

In  the  following  questions,  name  (1)  the  undefined  part,  (2)  the 
whole  quantity,  (3)  the  given  part. 

1.  A  merchant  sold  246  yd.  from  a  piece  of  cloth  258  yd.  in 
length.     How  many  yd.  had  he  remaining  ? 

2.  A  person  deposited  in  a  bank  $8495,  but  shortly  after  drew 
out  $1035.     How  much  had  he  left  in  the  bank  ?    If  $10  is  the 
unit  of  value  instead  of  $  1,  what  number  expresses  the  amount 
left  in  the  bank  ? 

3.  On  Tuesday  a  merchant  deposited   in  a  bank   $3475,  on 
Wednesday  $4690.     If  he  withdrew  $1010  on  Thursday,  how 
much  did  he  still  have  on  deposit  ? 

4.  What  is  the  difference  between  1  yr.  and  213  da.? 

5.  A  bankrupt  has  debts  amounting  to  $8496;  his  assets  are 
$  3015.     How  much  more  does  he  owe  than  he  can  pay  ? 

6.  A  man  left  property  to  the  value  of  $36,875  to  his  two 
children.     The  son  received  $14,250;   what  was  the  daughter's 
share  ? 

7.  At  an  election  the  successful  candidate  received  953  votes, 
and  the  unsuccessful  candidate  613  votes.     Find  the  majority  of 
the  former. 

44.    Computers'  Method. 

From  94,275  take  67,492.           Thus :  2  and  3  are  5  ;  9  and  8,  17  ; 

9497^  carry  1  to  4  as  in  addition,  making  it  5  ; 

'  5  and  7  are  12  ;  carry  1  to  7,  making  it 

3 '  ^  8  ;  8  and  6  are  14;  carry  1  to  0,  making 

26783  it  7  ;  7  and  2  are  9. 

The  numbers  3,  8,  7,  G,  and  2  are  written  down  in  order  to  give  the 
remainder. 


SUBTRACTION 


37 


45.   Find  the  difference  between  642  and  375. 


642 
375 


267 


As  we  cannot  take  5  units  from  2  units,  we  take  1  ten  from  the  4 
tens,  and  adding  this  1  ten,  which  equals  10  units,  to  the  2  units,  we 
have  12  units.  Then  5  units  from  12  units  leaves  7  units,  which  we 
write  under  the  units'  column.  Now  as  we  took  1  ten  from  4  tens, 
we  have  left  only  3  tens  ;  we  borrow  1  hundred  from  the  6  hundreds, 


and  considering  the  1  hundred  as  10  tens,  we  add  it  to  the  3  tens,  making  13 
tens  ;  then  7  tens  from  13  tens  leaves  6  tens,  which  we  write  under  the  tens' 
column. 

Now  as  we  took  1  hundred  from  6  hundreds,  we  have  left  only  5  hundreds ; 
hence  we  subtract  3  hundreds  from  5  hundreds,  leaving  only  2  hundreds, 
which  we  write  in  the  hundreds'  column. 

The  remainder,  or  difference,  is  thus  2  hundreds,  6  tens,  and  7  units, 
or  267. 


Exercise  19 


In  the  following  questions  prove  the  correctness  of  your  results 
by  adding  the  two  parts  and  finding  the  sum  equal  to  the  whole 
quantity. 


1. 


4. 


7. 


10. 


16. 


$653 
269 

921  min. 
87  « 

3849  yr. 
2567' 


« 


8000  yd. 
5348  « 


13  43970  pt. 

26784  « 


34060  bu. 
29143  " 


2. 


5. 


8. 


11. 


14. 


17. 


19. 


850439  T. 
473642  « 


307  dimes 
268   " 

255  hr. 
99  « 

9345  in. 

8367  " 

9041  rd. 
7385  « 

50062  qt. 
37891  " 

986403  oz. 

728547  " 


3. 


6. 


9. 


12. 


15. 


18. 


642  sec. 
375  « 

907  da. 
859  " 

7007  ft. 
6609  « 

7968  mi. 
2693  « 

12009  gal. 
11376  " 

620703  Ib. 


20. 


444444 

759826  A. 
378934  « 


« 


38 


*  Subtract: 

1.  57261 

38877 

4.  89437 
15790 

7.  654375 
412884 

10.  233826 
204739 

13.  164326 
48476 


ARITHMETIC 
Exercise  20 

2.  40359 
9998 

5.  67182 
30293 

8.  986392 
826957 

11.  310865 
270326 

14  982623 
897674 


3.  10000 
1021 

6.  81349 
47538 

9.  303233 
192001 

12.  605487 
584598 

15.  1000101 

707707 


*  Subtract: 

1.  755903 
699004 

4.  100794 
81685 

7.  4731246 
4342760 

10.  3801572 
2003789 

13.  1217191 
1038182 

16.  5468305 
1490673 

19.  8235460 
3530089 


Exercise  21 

2.  640021 
400569 

5.  143812 
109758 

8.  9487352 
5999999 

11.  5745861 
2837154 

14.  4100293 


192586 


17,  7086543 
2889454 

20.  2679953 
1346397 


3.  716287 
662763 

6.  948735 
473596 

9.  1737682 
739908 

12.  5048650 
4243091 

15.  2047000 
1054888 

18.  1671498 
536819 

21.  1521815 
1432568 


*  Use  computers1  method  of  subtraction. 


SUBTRACTION  39 

Exercise  22 

1.  Subtract  231  cu.  in.  from  2772  cu.  in.,  and  from  the  remain- 
der, and  so  on,  until  no  remainder  is  left.     If  1  gal.  contains 
231  cu.  in.,  how  many  gal.  are  there  in  2772  cu.  in.  ? 

2.  Subtract  320  rd.  from  2880  rd.,  and  from  each  remainder 
until  none  is  left.    If  the  unit  of  length,  1  mi.,  is  equal  to  320  rd., 
how  many  such  units  are  there  in  2880  rd.  ? 

3.  Subtract  1760  yd.  from  15,840  yd.,  and  from  each  remain- 
der until  none  is  left.     If  1  mi.  contains  1760  yd.,  how  many  ini. 
are  there  in  15,840  yd.  ? 

4.  Subtract,  as  in  question  3,  5280  ft.  from  68,640  ft.     If  1 
mi.  is  equal  to  5280  ft.,  how  many  mi.  are  equal  to  68,640  ft.  ? 

5.  Subtract,  as  in  question  3,  144  sq.  in.  from  864  sq.  in.         If 
1  sq.  ft.  contains  144  sq.  in.,  how  many  sq.  ft.  are  there  in  864 
sq.  in.  ? 

6.  Subtract,  as  in  question  3,  4840  sq.  yd.  from  53,240  sq.  yd. 
If  1   A.  contains  4840  sq.  yd.,   what  is   the   number   of  A.   in 
53,240  sq.  yd.  ? 

7.  Subtract,  as  in  question  3,  1728  cu.  in.  from  15,552  cu.  in. 
If  1  cu.  ft.  contains  1728  cu.  in.,  how  many  cu.  ft.  are  there  in 
15,552  cu.  in.  ? 

8.  Subtract,  as  in  question  3,  365  da.  3  times,  and  366  da.  once, 
from  2922  da.     How  many  yr.  are  there  in  2922  da.  ? 

Exercise  23 

Solve  the  following  questions  and  verify  your  answers. 

1.  Subtract  $819  from  $918,  explaining  the  process. 

2.  A  speculator  sold  cattle  at  a  loss  of  $3145  and  some  horses 
at  a  gain  of  $  2578.    How  much  did  he  lose  on  both  transactions  ? 

3.  A  merchant  exchanges  a  stock  of  goods  worth  $  6725,  and 
a  house  worth  $  3120,  with  a  farmer  for  a  farm  valued  at  $  5900, 


40  ARITHMETIC 

the  farmer  paying  the  balance  in  money.     What  sum  must  the 
merchant  receive  ? 

4.  If  the  measured  quantity  be  6743  bbl.  of  sugar,  and  one 
of  the  parts  is  1987  bbl.,  find  the  other  part. 

5.  Make  and  solve  a  question  in  which  the  whole  quantity 
and  one  part  are  given. 

6.  A  lends  B  $9780;  B  repays  A  by  giving  him  bank  stock 
to  the  amount  of  $  1946,  a  farm  worth  $  6385,  and  the  balance  in 
cash.     How  much  cash  did  B  pay  A  ? 

7.  A  is  worth  $6215,  B  is  worth  $876  less  than  A,  and  C  is 
worth  as  much  as  A  and  B  together,  lacking  $  2343.     How  much 
are  B  and  C  worth,  respectively  ?    How  much  are  all  three  worth  ? 

8.  How  much  larger  is  Lake  Erie  than  Lake  Ontario  ?    Lake 
Superior  than  Lake  Michigan  ?     The  total  areas  of  the  three 
smaller  lakes  than  Lake  Superior  ?     (For  the  areas  of  these  lakes 
see  Ex.  13,  question  6,  in  Addition.) 

If  100   sq.    mi.    is   the   unit,   what    number    expresses    these 
differences  ? 

9.  What  is  the  difference  between  643  and  579  when  the  unit 
of  value  is  $1?   $10?    $100?    $1000?    $10,000?    $100,000? 
$  1,000,000  ? 

10.  How  much  greater  are  253  units  of  $  1000  than  1864  units 
of  $  100  ?     What  number  expresses  the  difference  when  $  100  is 
the  unit  of  measure  ?     When  $  10  is  the  unit  of  measure  ? 

11.  A  man  bought  a  house  and  lot  for  $8450.     He  spent 
$  1379  in  improvements  and  $  212  for  insurance.     He  then  sold 
the  house  and  lot  for  $12,000;    did  he  gain  or  lose,  and  how 
much? 

12.  A  collector  received  $1300  from  five  men;  from  the  first 
he  received  $  367,  from  the  second  $  194  less  than  from  the  first, 
from  the  third  $  36  more  than  from  the  second,  from  the  fourth 
as  much  as  from  the  second  and  third  together.     How  much  did 
he  collect  from  the  fifth  ? 


SUBTRACTION  41 

13.  From    the    difference    between   784   and   8395,   take   the 
difference  between  17,012  and  21,410. 

14.  Two  men   start  from  the  same  point  and  travel  in  the 
same   direction.     The  first  travels  84  mi.   in  one  day  and  the 
second  69  mi.     How  far  were  they  apart  at  the  end  of  the  first 
day  ?     If  they  had  travelled  in  opposite  directions,  how  far  would 
they  have  been  apart  ? 

15.  The  population  of  Texas  in  1890  was  2,235,523,  and  of 
Illinois,  3,82(3,351.      How  much  greater  was  the  population  of 
Illinois  in  1890  than  that  of  Texas  ?     What  is  the  unit  in  this 
question  ? 

16.  The  length  of  the  St.  Lawrence  River  is  2000  mi.,  and 
the  area  drained   by  it  is  350,000  sq.  mi.      The  length  of   the 
Amazon   is   4000  mi.,  and  the  area  drained   by  it   is   2,500,000 
sq.  mi.     What  is  the  ratio  of  the  length  of  the  Amazon  to  that 
of   the   St.    Lawrence  ?     Show   by  subtracting   350,000   sq.    mi. 
successively  from  2,500,000  sq.  mi.,  how  many  times  greater  is 
the  area  drained  by  the  Amazon  than  that  drained  by  the  St. 
Lawrence,  and  find  the  remainder. 

17.  The  area  of  Texas  is  265,780  sq.  mi.,  of  England,  50,800 
sq.  mi.,  and  of   Germany,   208,700    sq.    mi.     How  much  larger 
is  Texas  than  the  united  area  of  England  and  Germany? 

18.  The  population  of  New  York  State  in  1870  was  4,387,464, 
in  1880  it  was  5,082,871,  and  in  1890  it  was  5,997,853.     What 
was  the  increase  in  population  from  1870  to  1880  ?     From  1880 
to  1890  ?     How  much  greater  was  the  latter  increase  than  the 
former  ? 

19.  If  the  colored  population  of  South  Carolina  in  1890  was 
689,141,  and  the  total  population  1,151,149,  how  much  larger 
was  the  colored  population  in  1890  than  the  white  population  ? 

20.  The  area  of  Texas  is  265,780  sq.  mi.,  of  Illinois  56,650 
sq.  mi.,  of  Oklahoma  39,030  sq.  mi.,  and  of  the  District  of  Co- 
lumbia 70  sq.  mi.     If  10  sq.  mi.  be  cut  off  of  Texas  for  an  Indian 


42  ARITHMETIC 

Reservation,  into  how  many  states  can  the  remainder  be  divided, 
making  as  many  as  possible  of  the  size  of  Illinois,  and  then  of 
Oklahoma,  and  of  the  District  of  Columbia  ? 

46.  From  25.3846  take  18.6397. 

o£  QQ/i«  ^e  wr^e  units  under  units,  tenths  under  tenths,  and  so 

on.     Beginning  at  the  right,  we  subtract  as  if  the  figures  were 

18.6397        integers,  and  place  the  decimal  point  in  the  difference  between 


74.4Q  units'  an(^  the  tenths'  column. 

Do  this  problem  by  the  computers'  method. 


Exercise  24 

Find  the  difference  : 

1.   26.437  2.    94.568  3.   102.4951 

15.254  29.783  58.2876 


From: 

4.  75.093  take  34.267. 

5.  6.4297  take  3.5824. 

6.  41.7453  take  27.937. 

7.  3.1111  take  1.4682. 

8.  3.1416  take  .9885. 

9.  A  car  containing  24.875  T.  of  coal  was  divided  between 
A  and  B.     A  received  11.375  T.  ;  what  did  B  get  ? 

10.    Show  by  successive  subtractions  that  a  field  containing 
52.584  A.  can  be  divided  into  6  fields,  each  containing  8.764  A. 


CHAPTER   V 

MULTIPLICATION 

47.   1.    Beginning  with  $2,  add  by  $2,  till  you  reach  $26. 
What  are  the  $  2  called?     Addends.     What  the  result  ?     Sum. 

2.  In  getting  this  sum  have  you  definitely  thought  of   how 
many  $  2  there  are  ?     JVo.     Do  you  know  from  the   sum   how 
many  there  are  ?     No. 

3.  If  you  add  $  2  to  $  2,  etc.,  till  you  reach  the  sum,  $  182,  do 
you  know  how  many  twos  there  are  ?     ^Vb. 

4.  How  do  you  look  upon  the  sum  $26  (say)  and  the  $2? 
The  $  26  is  simply  the  sum  of  an  unknown  number  of  $  2. 

5 .  Now  count  the  number  of  $  2.     There  are  13.     Did  you 
think  of  this  13  in  the  addition  process  ?     No. 

6.  Now  consider  this  13  in  relation  to  the  addend  $2,  and  the 
sum  $  26,  what  new  idea  is  introduced  ?     The  idea  of  hoiv  many 
times  $  2  is  repeated  to  make  $  26? 

7.  Then  what  is  the  number  which  measures  $  26  ?    13.    What 
is  the  unit  of  measure  ?     $  2.     From  what  you  know  of  number 
say  what  ratio  13  is  ?     The  ratio  of  $  26  to  $  2.     We  say  at  once 
(without  adding)  that  13  times  $  2  is  $  26. 

8.  In  this  do  we  depend  at  all  on  addition?     Yes.     We  first 
find  the  sum,  and  connect  this  in  memory  with  the  number  of 
times  the  addend  is  repeated. 

9.  But  is  it  then  correct  to  say  that  the  processes  $  2  +  $  2  + 
$  2  . . •  =  $  26,  is  identical  with  the  process  13  x  $  2  =  $  26  ?     No; 

43 


44  ARITHMETIC 

for  13  represents  the  "  new  idea '  referred  to,  and  $  2  has 
become  a  definite  unit  of  measure,  which  with  13  denotes  the 
quantity  $  26.  The  addend  has  become  a  factor,  and  the  sum 
a  product. 

48.  Find  the  cost  of  9  yd.  of  cloth  at  15  a  yd. 

(1)  Here  we  think  of  $5  as  a  derived  unit  measuring  the  value  of  1  yd. 
Hence  the  cost  of  9  yd.  is  equal  to  9  x  $  5,  or  to  $  45. 

(2)  Thus  45,  the  number  of  primary  units  in  the  total  cost,  is  called  the 
product  of  the  number  of  primary  units  in  the  derived  unit  $  5,  which  is  5, 
by  the  number  of  units,  viz.  9,  in  the  given  quantity  of  cloth. 

(3)  In  the  above  example  the  total  cost  was  given  by. 9  units  of  $5  each, 
and  after  multiplication  by  45  units  of  $  1  each.     Thus  multiplication  does 
not  change  the  total  cost  (i.e.  the  measured  quantity);  it  changes  only  the 
number  which  measures  it  (in  this  case  from  9  to  45)  by  changing  the  unit 
of  measure,  $5,  to  the  primary  unit,  $  1. 

The  numbers  to  be  multiplied  together,  viz.  9  and  5,  are  called  factors  of 
the  product,  i.e.  of  the  number  that  measures  the  quantity. 

49.  Multiplication  is  the  operation  of  finding  the  number 
of  primary  units  in  a  quantity  expressed  by  a  given  number 
of  derived  units,  or,  more  briefly, 

Multiplication  is  the  operation  of  finding  the  product  of 
two  numbers. 

The  Multiplicand  is  the  derived  unit  of  measure. 

The  Multiplier  denotes  ho\v  many  times  this  unit  of  meas- 
ure is  to  be  repeated,  i.e.  it  denotes  the  ratio  of  the  measured 
quantity  to  the  unit  of  measure. 

50.  8  x  16  is  read  8  times  $6. 

$6  x  8  is  read  $6  multiplied  by  8. 
x  is  the  Sign  of  Multiplication. 


MULTIPLICATION 


45 


MULTIPLICATION  TABLE 


Twice 

Three  times 

Four  times 

Five  times 

Six  times 

Seven  times 

1    is     2 

1   is     3 

1    is     4 

1    is     5 

1   is     6 

1   is     7 

2    "     4 

2    "     6 

2    "     8 

2    "    10 

2    "    12 

2    "   14 

3    "     6 

3    "     9 

3    "    12 

3    "    15 

3    "    18 

3    "   21 

4    "     8 

4    «    12 

4    "    16 

4    »   20 

4    "   24 

4    »   28 

5    "   10 

5    "    15 

5    "   20 

5    "   25 

5    "   30 

5    "    35 

6    "   12 

6    "    18 

6    "   24 

6    "   30 

6    "   36 

6    "   42 

7    "   14 

7    "   21 

7    "   28 

7    "    35 

7    "   42 

7    "   49 

8    »   16 

8    "   24 

8    "   32 

8    "   40 

8    "   48 

8    "   56 

9    "   18 

9    "   27 

9    "   30 

9    "    45 

9    "   54 

9    "   63 

10    "   20 

10    "   30 

10    "   40 

10    "    50 

10    "   60 

10    "   70 

11    "   22 

11    "   33 

11    "   44 

11    "   55 

11    "   66 

11    "    77 

12    «   24 

12    "   36 

12    "   48 

12    "   60 

12    "    72 

12    "   84 

Eight  times 

Nine  times 

Ten  times 

Eleven  times 

Twelve  times 

1    is     8 

1   is       9 

1   is     10 

1   is     11 

1    is     12 

2    "    16 

2    "     18 

2    "     20 

2    "     22 

2    "     24 

3    «  24 

3    "     27 

3    "     30 

3    "     33 

3    "     36 

4    "   32 

4    "     36 

4    "     40 

4    "     44 

4    "     48 

5    "   40 

5    "     45 

5    "     50 

5    "     55 

5    "     60 

6    «   48 

6    »     54 

6    »     60 

6    "     66 

6    "     72 

7    "    56 

7    "     63 

7    «     70 

7    "     77 

7    u     84 

8    "   64 

8    "     72 

8    "     80 

8    "     88 

8    "     96 

9    "   72 

9    "     81 

9    "     90 

9    "     99 

9    "    108 

10    "   80 

10    "     90 

10    "    100 

10    "    110 

10    "    120 

11    "   88 

11    u     99 

11    "    110 

11    "    121 

11    "   132 

12    "   96 

12    "    108 

12    "    120 

12    "    132 

12    "   144 

46  ARITHMETIC 

51.  Develop  at  least  a  portion  of  the  multiplication  table  by  measuring. 
Let  a  line  be  drawn  on  the  board  at  least  36  in.  long,  and  let  12  strips  of 
paper  be  cut,  respectively,  1  in.,  2  in.,  3  in.,  •••,  12  in.  long,  representing 
different  units  of  measure. 

To  develop,  for  instance,  the  table  of  3's,  measure  along  the  line  3 
times  with  each  of  these  measures.  Then  with  a  yard  ruler  divided  into 
inches  measure  each  result  in  turn. 

Hence  3  x  1  in.  =  3  in. 

3  x  2  in.  =  6  in. 
3  x  3  in.  =  9  in. 
etc.  =  etc. 
3  x  12  in.  =  36  in. 


i 

52.  From  the  above  diagram  it  is  evident  that  32  is  equal 
to  4  x  8,  or  8x4. 

Arrange  the  dots  to  show  that  32  is  equal  to  2  x  16,  or 
16  x  2. 

How  often  is  32  measured  by  4  ?  8  ?  2  ?  16  ? 

Exercise  25 

1.  Arrange  dots,  representing  any  units,  to  show  that  28  =  4  x  7, 
or  7  x  4. 

2.  Give  the  factors  of  45  (9  x  5  or  5  x  9),  66,  56,  72,  96,  63, 
90,  54,  99,  84,  132,  108. 

3.  Give  the  factors  of  9,  16,  25,  36,  49,  64,  81,  100,  121,  144. 

4.  Arrange  30  dots  to  show  how  often  30  is  measured  by  10. 
By  3. 

5.  How  often  is  72  measured  by9?8?6?12?4?18?3? 
24?  2?  36? 

6.  If  one  factor  of  96  is  12,  what  is  the  other  ?     If  one  factor 
is  8,  what  is  the  other  ? 


MULTIPLICATION 


47 


7.  What  will  5  yd.  of  cloth  cost  at  $4  a  yd.  ?     What  will 
4  yd.  cost  at  $  5  a  yd.  ? 

8.  If  9  men  can  do  a  piece  of  work  in  6  da.,  how  long  will  it 
take  1  man  to  do  it  ?     If  6  men  can  do  a  piece  of  work  in  9  da., 
how  long  will  it  take  1  man  to  do  it  ? 


53.   Suggestions  regarding  the  Multiplication  Table. 

1.  The  tables  of  10  and  11  present  no  difficulty,  the  product  being 
directly  associated  with  the  digit. 

2.  The  table  of  9  can  be  similarly  remembered.     The  product  is  made 
up  of  tens  and  units.     In  the  table  (up  to  10  x  9)  the  tens'  digit  is  always 
one  less  than  the  number  multiplied.     The  sum  of  the  digits  is  9. 

3.  In  the  table  of  5,  the  unit  digit  is  5  for  odd  multiplicands,  and  0  for 
even  ones. 

4.  By  association  ;  thus  if  6  x  9  =  54,  then  9  x  6  =  54. 

5.  Require  the  table  to  be  memorized  in  regular  order ;  also,  so  that  it 
can  be  given  by  the  pupil  in  irregular  order,  thus :  9  x  4  =  36,  9  x  6  =  54, 
9  x  10  =  90,  etc. 

6.  Drill,  requiring  instantaneous  oral  and  written  answers  to  such  ques- 
tions as  :  What  is  6x7?    9x8?     8x9? 

7.  Drill,  requiring  instantaneous  answers :    What  is  6x7  +  4?  9x8  +  3? 
8x9  +  7? 

8.  Extend  the  table  thus  :  5  x  70  =  ?     7  x  800  =  ?    4  x  120  =  ? 

9.  Drill  thus  :  42  +  7  =  ?     84  -  12  =  ?     54  -  8  =  ? 

10.  Give  two  factors, 
each  less  than  12,  of  36, 
54,  etc. 

11.  What  part  of  28 
is  12,  16  ?     Of  36  is  20, 
28  ?    (using  the  table  of 

4  as  a  basis) .  • 

12.  What  is  the  quan- 
tity whose   ratio   to  the 
unit  $9  is  equal  to  6  ? 

13.  How  many  in.  in 

5  ft.  4  in.  ?     How  many 

da.  in  4  wk.  6  da.  ?  Scale .-  i  m.  =  i  in, 


48 


ARITHMETIC 


54.  (1)  Find  the  area  of  an  oblong  5  in.  long  and  3  in. 
wide. 

Let  the  oblong  be  divided  into  3  strips  by  lines  1  in.  apart,  as  in  the 

figure. 

The  area  of  1  strip        =  5  sq.  in. 
/.  the  area  of  the  oblong  =  3x5  sq.  in. 

=  15  sq.  in. 

In  this  figure  there  are  two  units  of  measurement.  The  smaller  or  pri- 
mary unit  of  area  is  1  sq.  in.,  and  is  repeated  5  times  to  make  the  larger 
unit,  or  the  strip  ;  the  larger  or  derived  unit  of  one  strip  is  5  sq.  in.  and  is 
repeated  3  times  to  make  the  oblong  which  is  now  measured.  Make  a 
rectangle  5  in.  long  and  3  in.  wide.  Divide  it  as  in  the  figure  and  make  a 
mental  picture  of  the  resulting  figure. 

(2)  Reduce  9  pk.  6  qt.  to  qt. 

9  pk.  =  9  x  8  qt.  =  72  qt. 
/.  9  pk.  6  qt.  =  78  qt. 
Here  the  problem  is  to  add  6  qt.  to  9  units  of  8  qt.  each. 


in.  =  I  in. 


(3)  Find   the   volume  of  a  rectangular  solid   5  in.  long, 
4  in.  wide,  and  3  in.  thick. 

Let  the  solid  be  divided  into  3  slices  by  horizontal  planes  1  in.  apart. 
Let  the  upper  slice  be  divided  into  5  rows  by  vertical  planes  1  in.  apart. 


MULTIPLICATION  49 

Let  the  right-hand  row  be  divided  into  4  cu.  in.  by  vertical  planes  1  in. 
apart. 

The  volume  of  1  row  =  4  cu.  in. 
The  volume  of  5  rows  or  1  slice  =  5x4  cu.  in. 
The  volume  of  3  slices  or  the  solid  =  3  x  5  x  4  cu.  in. 

=  60  cu.  in. 

In  this  solid,  the  three  units  of  volume,  in  order  of  size,  are  the  primary 
unit  or  1  cu.  in.,  and  the  derived  units,  i.e.  the  rows  or  4  cu.  in.,  and  the 
slice  or  5x4,  i.e.  20  cu.  in. 

The  solid  is  made  up  of  how  many  units  of  each  kind  ?  Each  unit  is 
made  up  of  how  many  of  the  next  smaller? 

Exercise  26 

In  the  following  exercise,  where  possible,  make  drawings  and 
mental  pictures : 

1.  Find  the  area  of  the  following  oblongs,  draw  the  figure, 
and  name  the  primary  and  derived  units  of  area:  4  in.  by  6  in.; 
7  in.  by  9  in. ;  8  in.  by  11  in. ;  9  in.  by  12  in.     What  is  the  ratio 
of  the  area  of  the  oblong  to  the  primary  unit  ?     To  the  derived 
unit  ?     What  is  the  ratio  of  the  derived  to  the  primary  unit  ? 

2.  Find   the  number  of  sq.  ft.   in  a  sq.  yd.  ;  of  sq.  in.  in  a 
sq.  ft. 

3.  Find  the  area  of  the  floors  of  the  following  rooms  whose 
dimensions  are:   6  yd.,  7  yd.;  6  yd.,  8  yd.;  8  yd.,  9  yd.;  11  yd., 
12  yd. 

4.  Find  the  area  of  squares  whose  sides  are:  1,  2,  3,  4,  5,  6, 
7,  8,  9,  10,  11,  and  12  in.,  respectively. 

5.  Find  the  area  of  a  room  made  up  of  8  units  of  9  sq.  yd. 

6.  If  the  unit  of  length  is  4  ft.,  what  is  the  unit  of  area  ? 

7.  What  is  the  volume  whose  smallest  unit,  1  cu.  in.,  is  re- 
peated 3  times  to  make  the  next  larger,  this  6  times  to  make  the 
next  larger,  and  this  8  times  to  make  the  volume  ? 

8.  Find  the  volumes  of  rectangular  solids  whose  dimensions 
are :  2,  3,  and  4  in. ;  2,  4,  and  9  in. ;  3,  4,  and  7  in. ;  2,  5,  and  9 

E 


50  ARITHMETIC 

in.     What  are  the  volumes  of  the  primary  and   of   the  derived 
units  of  volume  ? 

9.    Find  the  number  of  cu.  ft.  in  a  cu.  yd. 

10.  If  4  in.  is  the  unit  of  length,  what  is  the  unit  of  volume  ? 

11.  Find  the  volumes  of  the  cubes  whose  sides  are  respec- 
tively 1,  2,  3,  4,  5,  6,  7,  8,  9,  10,  11,  12  in. 

12.  Reduce  to  lower  denominations: 

(1)  5  yd.  2  ft. ;  7  yd.  1  ft. ;  8  yd.  2  ft. ;  9  yd.  1  ft. ;  11  yd.  1  ft. ; 
12  yd.  2  ft. 

(2)  5  ft.  4  in. ;  8  ft.  10  in. ;  7  ft.  8  in. ;  11  ft.  3  in. ;  9  ft.  7  in. ; 
12  ft.  6  in. 

13.  3  sq.  yd.  5  sq.  ft. ;   5  sq.  yd.  7  sq.  ft. ;   12  sq.  yd.  6  sq.  ft. ; 

8  sq.  yd.  8  sq.  ft. ;  9  sq.  yd.  2  sq.  ft. ;  11  sq.  yd.  10  sq.  ft. 

14.  6  qt.  1  pt. ;  8  qt.  1  pt. ;  11  qt.  1  pt. ;  7  pk.  6  qt. ;  9  pk.  4  qt. ; 

9  bu.  3  pk. ;  8  bu.  2  pk. ;  11  bu.  3  pk. ;  7  gal.  2  qt. ;  9  gal.  3  qt. 

15.  7  wk.  4  da. ;  9  wk.  2  da. ;  11  wk.  6  da. ;  8  wk.  1  da. ;  12  wk. 

5  da. ;  5  hr.  40  min. ;  8  hr.  9  min. ;  9  hr.  22  min. ;  12  hr.  45  min. ; 

6  da.  4  hr. ;  8  da.  2  hr. 

16.  How  many  hr.  are  there  in  1  wk.  ? 

17.  The  perimeter  of  a  room  is  found  by  adding  twice  the 
width  to  twice  the  length.     Find  the  perimeter  of  rooms  whose 
dimensions  are :  6  yd.,  8  yd.  ;  7  yd.,  9  yd. ;  8  yd.,  11  yd. ;  11  ft., 
12  ft.;  13  ft.,  14  ft. 

55.    What  is  the  cost  of  6  town  lots  at  1894  a  lot? 

Here  we  think  of  the  whole  cost  as  6  units  of  $  894  each. 
EXPLANATION.  — The  unit  $894  may  be  considered  as  made  up  of  4  units 
of  one  dollar,  9  units  of  ten  dollars,  and  8  units  of  one  hundred  dollars. 

6x4  units  of  one  dollar  =  24  units  of  one  dollar  =  2  units  of 

<&  OQ4. 

ten  dollars  +  4  units  of  one  dollar. 

6          6x9  units  of  ten  dollars  =  54  units  of  ten  dollars. 
<m  coc?4          54  units  of  ten  dollars  +  2  units  of  ten  dollars  —  56  units  of  ten 

dollars  =  5  units  of  one  hundred  dollars  +  6  units  of  ten  dollars. 
6x8  units  of  one  hundred  dollars  =  48  units  of  one  hundred  dollars. 


MULTIPLICATION  51 

48  units  of  one  hundred  dollars  -f  5  units  of  one  hundred  dollars  =  53 
units  of  one  hundred  dollars  =  5  units  of  one  thousand  dollars  +  3  units  of 
one  hundred  dollars. 

5  units  of  one  thousand  dollars  +  3  units  of  one  hundred  dollars  +  6  units 
of  ten  dollars  +  4  units  of  one  dollar  =  $  5364. 

56.    The  method  generally  followed  by  the  pupil  is : 

6  times  4  =  24 ;  6  times  9  =  54  ;  54  and  2  =  56  ;  6  times  8  =  48  ;  48  and 
5  =  53. 

This  process  is  too  slow.  If  the  pupils  have  been  well  drilled  in  §  53, 
No.  7,  they  will  be  able  to  add  in  the  digit  to  be  carried  instantaneously,  and 
should  be  trained  thus  : 

6x4=  24;  6x9  =  56  (adding  in  the  2  in  one  process). 
6  x  8  =  53  (adding  in  the  5) . 

Exercise  27 

Multiply  separately : 

1 .  231  by  2,  4,  6,  8,  10,  and  12. 

2.  690  by  3,  5,  7,  9,  and  11. 

3.  897  by  4,  6,  8,  10,  and  12. 

4.  2463  by  3,  5,  7,  9,  and  11. 

5.  5781  by  2,  4,  8,  10,  and  12. 

6.  9654  by  3,  5,  7,  9,  and  11. 

7.  8267  by  2,  4,  6,  8,  10,  and  12. 

8.  5280  by  3,  5,  7,  9,  and  12. 

9.  1728  by  2,  4,  6,  8,  10,  and  12. 

10.  4840  by  3,  5,  7,  9,  and  11. 

11.  63,360  by  2,  4,  6,  8,  10,  and  12. 

12.  24,793  by  3,  5,  7,  9,  and  11. 

13.  98,654  by  2,  4,  6,  8,  10,  and  12. 

14.  89,743  by  3,  5,  7,  9,  and  11. 

15.  64,789  by  2,  4,  6,  8,  10,  and  12. 

16.  Solve  the  questions  in  Exercise  12  by  multiplication. 


52  ARITHMETIC 

Exercise  28 

1.  Find  the  perimeters,  and  also  the  areas  of  the  four  walls 
of  rooms  whose  dimensions  are : 

LENGTH  WIDTH  HEIGHT 

16  ft.  14  ft.  8  ft. 

17  «  15  «  8  « 
20  "  18  "  9  " 
22  «  20  "  12  " 

2.  A  cd.  of  wood  is  8  ft.  long,  4  ft.  wide,  4  ft.  high.     How 
many  cu.  ft.  does  it  contain  ? 

3.  Find  the  number  of  cu.  in.  in  a  cu.  ft. 

4.  A  gal.  of  water  will  exactly  fill  a  rectangular  box  11  in. 
long,  7  in.  wide,  and  3  in.  high.     Find  the  number  of  cu.  in.  in 
a  gal. 

5.  If  there  are  1760  yd.  in  1  mi.,  find  the  number  of  yd.  in 

2  mi. ;  5  mi. ;  8  mi. ;  9  mi.  ;   12  mi. 

6.  If  there  are  5280  ft.  in  1  mi.,  find  the  number  of  ft.  in 

3  mi. ;  6  mi. ;  7  mi. ;  9  mi. ;  11  mi. 

7.  Reduce  to  sq.  in. :  5  sq.  ft.;  8  sq.  ft.;  10  sq.  ft. ;  12  sq.  ft. 

8.  Reduce  to  sq.  ft. :  3547  sq.  yd.;  8426  sq.  yd.;  9819  sq.  yd. 

9.  If  1  sq.  mi.  contains  640  A.,  find  how  many  A.  there  are 
in  6  sq.  mi. ;  8  sq.  mi. ;  10  sq.  mi. ;  12  sq.  mi. 

10.  Reduce  to  cu.  in. :  4  cu.  ft. ;  5  cu.  ft. ;  7  cu.  ft. ;  9  cu.  ft. ; 

11  cu.  ft. 

11.  Reduce  to  da.  :  453  wk. ;  769  wk. ;  827  wk. ;  852  wk. 

12.  Reduce  to  qt. :  765  gal. ;  917  gal. ;  763  gal. ;  789  gal. 

13.  Reduce  to  qt.  :  735  pk. ;  892  pk. ;  679  pk. ;  728  pk. 

14.  Reduce   to    da. :    3  yr. ;    5   yr. ;    6   yr. ;    8  yr. ;    11    yr. ; 

12  yr.     (1  yr.  =  365  da.) 

15.  If  there  are  640  A.  in  1  sq.  mi.,  find  the  number  of  A. 
in  6  sq.  mi. ;  in  9  sq.  mi. ;  in  12  sq. 


MULTIPLICATION  53 

16.  Show  that  4  units  of  one  dollar  multiplied  by  7  tens  is 
equal  to  the  product  found  by  multiplying  4  units  of  ten  dollars 
by  7. 

17.  Show  that  9  units  of  ten  dollars  multiplied  by  7  tens  is 
equal  to  63  units  of  one  hundred  dollars. 

57.  If  the  number  expressing  the  ratio  of  the  measured 
quantity  to  the  unit  of  measure  ft  894  is  76,  what  is  the 
quantity  ? 


$  894          ^ie  exPlanation  is  similar  to  that  given  in  §  55.     Since,  when 
_  -,     7  is  used  as  a  multiplier,  the  4  units  of  one  dollar  are'  multiplied 
_    by  7  tens,  the  product  is  the  same  as  that  found  by  multiplying 
5364    4  units  of  ten  dollars  by  7.     This  is  28  units  of  ten  dollars,  and  is 
6^58       ecmal  to  2  units  of  one  hundred  dollars  and  8  units  of  ten  dollars. 
Hence  the  8  is  written  under  the  6  in  the  tens'  column,  and  the 
07944     2  is  carried  to  be  added  in  the  hundreds'  column,  and  so  on. 
To  prove  the  answer  correct,  multiply  76  by  894  ;  thus  : 

76 
894 
304 
684 
608 
.-.  the  answer  is  correct.  67944 

Exercise  29 

Multiply  and  prove  your  answers  correct  : 

1.  423  by  36.  11.  8647  by  365. 

2.  479  by  32.  12.  7245  by  168. 

3.  295  by  16.  13.  8939  by  224. 

4.  798  by  24.  14.  6558  by  144. 

5.  581  by  52.  15.  9275  by  231. 

6.  649  by  27.  16.  9475  by  1760. 

7.  959  by  24.  17.  8213  by  5280. 

8.  764  by  31.  18.  4781  by  1728. 

9.  953  by  56.  19.  5893  by  2240. 
10.  825  by  48.  20.  6439  by  1728. 


54  ARITHMETIC 

Exercise  30 
Multiply : 

1.  8245  by  684.  6.  8746  by  675. 

2.  7639  by  797.  7.  9687  by  897. 

3.  5927  by  395.  8.  4786  by  478. 

4.  4399  by  927.  9.  9467  by  769. 

5.  8999  by  868.  10.  8769  by  567. 

Exercise  31 

In  the  following  questions  state  which  is  the  unit  of  measure 
and  which  is  the  number : 

1.  If  1  mi.  contains  320  rd.,  find  the  number  of  rd.  in  2897  mi. 

2.  If  1  mi.  contains  1760  yd.,  find  the  number  of  yd.  in  1679  mi. 

3.  If  1  mi.  contains  5280  ft.,  find  the  number  of  ft.  in  834  mi. 

4.  If  1  sq.  ft.  contains  144  sq.  in.,  find  the  number  of  sq.  in.  in 
a  rectangle  27  ft.  long  and  18  ft.  wide. 

5.  Find  the  number  of  sq.  ft.  in  a  garden,  the  shape  of  an 
oblong,  which  is  16  yd.  long  and  14  yd.  wide. 

6.  If  1  sq.  mi.  contains  640  A.,  how  many  A.  are  there  in  a 
township  containing  36  sq.  mi.  ? 

7 .  Find  the  value  of  6  units  of  land  at  $  85  a  unit. 

8.  If  1  in.  be  taken  as  the  unit  of  length,  how  many  units  of 
area  are  there  in  the  surface  of   a  box  whose  dimensions  are 
respectively  2,  3,  and  4  of  the  next  higher  unit  of  length  ? 

9.  If  1  ft.  be  taken  as  the  primary  unit  of  length,  what  is 
the  number  of  the  primary  units  of  volume  in  a  rectangular  solid 
whose  dimensions  contain  4,  5,  and  6  of  the  next  higher  unit  of 
length? 


MULTIPLICATION  55 

10.  If  the  unit  of  length  is  2  ft.,  what  is  the  unit  of  volume? 
How  many  cu.  ft.  are  there  in  the  volume  of  a  rectangular  solid 
whose   dimensions   contain  respectively  3,  4,   and  5   such  units 
of  length  ? 

11.  What  is  the  area  of  the  surface  of  this  solid  in  square 
feet? 

12.  If  1   cu.  ft.  contains  1728  cu.  in.,  find  the   number  of  cu. 
in.  in  a  rectangular  solid  8  ft.  long,  4  ft.  wide,  and  4  ft.  high. 

13.  If  1  cu.  yd.  contains  27  cu.  ft.,  find  the  number  of  cu.  ft. 
in  a  rectangular  solid  whose  dimensions  are  G,  4,  and  3  yd. 

14.  If  1  cd.  of  wood  contains  128  cu.  ft.,  how  many  cu.  ft.  are 
there  in  936  cd.  ? 

15.  If  1  gal.  contains  231  cu.  in.,  how  many  cu.  in.  are  there  in 
a  vessel  which  contains  888  gal.  ? 

16.  A  quantity  contains  the  unit  365  da.  896  times.     Find  the 
quantity. 

58.  (1)  A  drover  bought  36  horses  at  $145  a  head,  and  96 
cows  at  §28  a  head.     Which  cost  the  more,  and  how  much  ? 

Here  the  problem  is  to  find  the  difference  between  36  units  of  $  145  each 
and  96  units  of  $  28  each. 

(2)  A's  barn  cost  $175,  his  house  16  times  as  much,  and  his 
farm  cost  as  much  as  both.     What  was  the  cost  of  all  ? 

Here  the  problem  is  to  find  the  cost  of  1  +  16  +  17,  or  34  units  of  $  175 
each. 

Exercise  32 

1.  Exemplify  the  truth  that  two  or  more  factors  will  give  the 
same  product  in  whatever  order  they  are  multiplied. 

2.  A  speculator  bought  150  head  of  cattle  and  47  mules.     He 
made  a  profit  of  $13  ahead  on  the  former  and  $17  each  on  the 
latter.     What  was  gained  by  the  speculation  ? 


56 


ARITHMETIC 


3.  A  ship  sailed  56  hr.  at  the  rate  of  11  mi.  per  hr.,  when  she 
encountered  a  storm  of  16  hours'  duration,  which  drove  her  back 
at  the  rate  of  14  mi.  per  hr.     How  far  from  port  was  she  at  the 
expiration  of  the  72  hr.  ? 

4.  A  is  worth  $  1265,  B  is  worth  4  times  as  much  as  A  and 
$  183,  and  C  is  worth  ,3  times  as  much  as  A  and  B  lacking  $2343. 
How  much  are  B   and  C  worth,  respectively  ?     How  much  are 
they  all  worth  ? 


Make  out  a  bill  for  the  following  goods: 
23  yd.  cotton  @  11^  ;    13  yd.  gingham  @  23?  ; 


5. 


25  yd.  flannel  @  37^  ;  18  yd.  tweed  @  $  1.50  ; 
12  yd.  serge  @  $  1.75;  6  yd.  broadcloth  @  $4.50. 

6.  A  produce  merchant  exchanged  48  bu.  of  oats  at  39^  per 
bu.  and  13  bbl.  of  apples  at  $3.85  a  bbl.  for  200  Ib.  of  butter  at 
37^  a  Ib.     How  much  should  he  pay  to  settle  the  account  ? 

7.  A  grain  dealer  buys  4795  bu.  of  wheat  in  Chicago  at  63^  a 
bu.,  and  ships  it  to  New  York  at  a  cost  of  3^  a  bu.     Find  his 
gain  if  he  sells  it  in  New  York  for  71^  a  bu. 

8.  A  man  bought  51  horses  at  $  97  each,  and  sold  them  at 
$  136  each.     How  much  did  he  gain  ? 

9.  Find  the  amount  of  the  following  bill  : 

63  brooms,  at  16^  each  ; 
13  yd.  print,  at  11^  per  yd.  ; 
17  Ib.  tea,  at  35^  per  Ib.  ; 
4  doz.  oranges,  at  4^  each  ; 
287  Ib.  sugar,  at  5^  per  Ib.  ; 
84  eggs,  at  13^  per  doz. 

10.  Bought  oranges  at  the  rate  of  18^  a  doz.,  and  sold  them 
at  the  rate  of  6  oranges  for  15^.  How  much  did  I  gain  on 
11  boxes,  each  containing  20  doz.? 


MULTIPLICATION  57 

11.  If  in  question  10  two  boxes  were  spoiled,  what  would  have 
been  the  gain  ? 

12.  A  grocer  buys  150  Ib.  of  coffee  at  23^  a  lb.,  and  39  Ib.  of 
chiccory  at  6^  a  lb.     He  pays  a  duty  of  2^  a  lb.  on  each,  and 
mixes  and  sells  the  mixture  at  33^  a  lb.     Find  his  profit. 

13.  A  book  contains  457  pages,  each  page  containing  39  lines, 
averaging  13  words  to  a  line.    Find -the  number  of  words  in  the 
book. 

14.  How  far  will  a  bicyclist  travel  in  27  da.,  if  he  travels  9  hr. 
a  da.  at  12  mi.  an  hr.  ? 

15.  Two  vessels  start  from  the  same  point  and  travel,  the  one 
down  a  river  at  the  rate  of  15  mi.  an  hr.,  the  other  up  the  river  at 
the  rate  of  9  mi.  an  hr.     How  far  will  they  be  apart  in  8  hr.  ? 

16.  If  the  first  vessel  travelled  up  the  river  at  the  rate  of  12  mi. 
an  hr.,  how  far  apart  would  they  be  in  8  hr.  ? 

17.  A  speculator  bought  45  A.  of  land  at  $  65  an  A.,  and  63  A. 
at  $  78  an  A.     If  he  sold  the  whole  at  $  75  an  A.,  how  much  did 
he  gain  or  lose  ? 

59.    Multiply  .948  by  6. 
.948 

948  thousandths  multiplied  by  6  equals   5688  thousandths   or 
5.688. 


5.688 

Exercise   33 

Multiply : 

1.  .5  by  9.  5.  .624  by  3. 

2.  .8  by  3.  6.  .842  by  9. 

3.  .26  by  3.  7.  .1416  by  25. 

4.  ,39  by  8.  8.  .988  by  76. 


58  ARITHMETIC 

9.    3.54  by  12.  12.    3.295  by  16. 

10.  .543  by  36.  13.    .7568  by  144. 

11.  4.79  by  32.  14.   8.754  by  172. 

15.  The  circumference  of  a  circle  is  3.1416  times  the  diameter. 
What  is  the  circumference  of  a  circle  whose  diameter  is  27  yd.  ? 

16.  Find  the  number  of  sq.  yd.  in  1  sq.  ch.,  there  being  30.25 
sq.  yd.  in  1  sq.  rd.,  and  16  sq.  rd.  in  1  sq.  ch. 

17.  A  cu.  ft.  contains  7.48  gal.  of  water.     Find  how  many  gal. 
can  be  poured  into  a  tin-lined  box,  whose  interior  dimensions  are 
3  by  4  by  6  ft. 

18.  Find  the  weight  of  a  rectangular  solid  of   oak,  3  ft.  by 
2  ft.  by  1  ft.,  weighing  47.375  Ib.  per  cu.  ft. 

19.  A  drover  bought  12   sheep   at   $  5.375  per  head,   36   at 
$4.625,  and  212  at  $4.125.     Find  the  total  cost. 

20.  Multiply  each  of  the  following  numbers  by  10  :  .43  ;  .576 ; 
4.23;  .017;  89.4263. 

21.  Write  down  the  product  of  each  of  the  following  numbers 
multiplied  by  10 :   7.4;  8.946;  5.32;  .008;  62.9347. 

22.  State  how  to  write  the  product  obtained  by  multiplying  a 
decimal  by  10. 

23.  Multiply  by  100  each  of  the  following:  .435;  8.027;  9.12; 
46.5928. 

24.  State  how  to  write  the  product  obtained  by  multiplying  a 
decimal  by  100.     By  1000.     By  10,000. 


CHAPTER    VI 
DIVISION 

60.  What  will  4  oranges  cost  at  5^  apiece  ?     If  4  oranges 
cost  20^,  what  is  the  cost  of  each?     What  must  50  be  multi- 
plied by  to  get  20^  ? 

At  $5  a  yd.,  how  much  will  9  yd.  of  cloth  cost?  What 
must  $3  be  multiplied  by  to  get  $27?  At  $3  a  yd.,  how 
many  yd.  can  be  bought  for  $  27  ? 

61.  Of  what  product  are  5  and  6  the  factors?  (30.)     If  5 
is  one  factor  of  30,  what  is  the  other?     Of  what  product  are 
12  and  4  the  factors  ?     If  4  is  one  factor  of  48,  what  is  the 
other?     If  6  is  one  factor  of  42,  what  is  the  other?     If  9  is 
one  factor  of  63,  what  is  the  other  ?     4  is  one  factor  of  each 
of  the  following  numbers ;  what  are  the  other  factors  ?     24. 
36,  20,  28,  and  16. 

62.  In  Multiplication  we  are  given  two  factors  and  we  are 
required  to  find  their  product. 

In  Division,  on  the  other  hand,  we  are  given  the  product, 
and  also  one  of  the  factors,  and  we  are  required  to  find  the 
other  factor. 

Thus:  Find  how  many  yd.  of  cloth  at  $5  a  yd.  can  be 
bought  for  |45? 

In  this  problem,  45,  the  number  measuring  the  cost  of  the 
cloth,  is  the  product  of  two  factors.  One  of  these  is  5,  the 
given  number,  which  measures  the  value  of  the  unit,  and 
the  other  is  9,  which  is  the  number  of  yards. 

59 


60  ARITHMETIC 

If  9  yd.  of  cloth  cost  $45,  what  will  1  yd.  cost? 
As  before,  45  is  the  product  of  two  factors.     The  given  fac- 
tor is  9,  and  the  required  factor,  5.     Therefore  1  yd.  costs  $  5. 

63.  Division  is  the  operation  of  rinding  either  of  two  fac- 
tors, when  their  product  and  the  other  factor  are  given. 

The  factor  found  is  called  the  Quotient.  It  shows  how 
often  the  Divisor  is  contained  in  the  Dividend. 

The  given  factor  is  called  the  Divisor. 

The  given  product  of  the  Quotient  and  Divisor  is  called 
the  Dividend. 

When  the  Divisor  is  not  contained  an  exact  number  of 
times,  the  excess  is  called  the  Remainder.  See  §  71. 

64.  The  Sign  of  Division  is  -^  ;  thus  $8  -5-  $2  =  4  is  read 
1 8  divided  by  I  2  is  equal  to  4. 

$  8  -:-  2  =  1 4,  is  read  $  8  divided  by  2  is  equal  to  $  4. 
9  -5-  3  may  also  be  written  |,  where  9  is  the  dividend  and  8 
the  divisor. 

65.  When  the  divisor  does  not  exceed  12,  the  operation 
can  be  performed  mentally,  and  the  process  is  called  Short 
Division. 

When  all  the  different  steps  of  the  division  are  written,  the 
process  is  called  Long  Division. 

66.  SUGGESTIONS    TO    THE    TEACHER.  -  -  Give    questions 
similar  to  the  following,  in    order   to   secure   facility  in   in- 
terpreting results   and  accuracy  and  rapidity  in    using    the 
multiplication  table. 

Thus  :  (1)  A  product  is  72  ;  one  factor  is  8.    Find  the  other. 

(2)  What  is  $36 -4?     $72-$9?     |72-9?    132-11? 
Associate  simple  practical  questions  with  these  numbers. 

(3)  Extend  the  table  thus:  Divide  210  by  7;  3500  by  5; 
450  by  90 ;  840  by  12. 


DIVISION 


61 


(4)  Give  the  quotient  and  remainder  when  86  is  divided 
by  7  ;  93  by  12  ;  43  by  6. 

(5)  Reduce  to  the  next  higher  denomination  :  32  qt. ;  40^ ; 
96  in. ;  45  da. ;  450  min. 

(6)  The  unit  of  area  is  9  sq.  rd.     What  number  expresses 
the  ratio  of  the  area  of  a  field  containing  270  sq.  rd.  to  the 
unit  of  area  ? 

67.  If  1  T.  of  coal  costs  16,  how  many  T.  will  14764  buy? 

Here  the  product  is  $4764  ;  one  factor  is  $6,  and  we  are  required  to  find 
the  other  factor,  which  is  the  number  of  T. 

6)4764 
794 

6  divides  47  of  the  hundreds'  unit  7  times  in  the  hundreds'  place,  with  a 
remainder  5  of  the  hundreds'  unit ;  5  of  the  hundreds'  unit  and  6  of  the 
tens'  unit  equal  56  of  the  tens'  unit. 

6  divides  56  of  the  tens'  unit  9  times  in  the  tens'  place,  with  a  remainder 
2  of  the  tens'  unit.  2  of  the  tens'  unit  and  4  of  the  one-unit,  equal  24,  which 
divided  by  6  equals  4. 

/.  the  number  of  T.  =  794. 

68.  (1)  Find  the  length  of  an  oblong  which  contains  96 
sq.  in.  and  is  8  in.  wide. 


co 


GO 


96  SQ.  IN. 


o 
+-> 

c 


(0 

o 

to 


62  ARITHMETIC 

Cut  off  from  the  oblong  a  strip  1  ft.  wide.     This  strip  contains  8  sq.  in. 
Here  we  are  given  the  whole  quantity,  or  96  sq.  in.,  and  the  measuring  unit, 
8  sq.  in.     The  number  12  gives  the  number  of  primary  units  of  1  in.  con- 
tained in  the  length.     Therefore  the  length  is  12  in. 

The  area  of  1  strip  1  in.  wide  =  8  sq.  in. 
The  number  of  strips  1  in.  wide  =  96  sq.  in.  -r-  8  sq.  in.  =  12. 

.-.  the  length  =  12  in. 

(2)  Find  the  number  of  yd.  of  carpet  required  to  carpet 
a   room   32   ft.   long   and  26   ft.  wide,  the   carpet   running 
lengthwise,  if  each  strip  is  2  ft.  wide. 

The  number  of  strips  of  carpet  =  26  ft.  -4-  2  ft.  =  13. 
The  length  of  the  carpet  =  13  x  32  ft.  =  416  ft. 

=  138  yd.  2  ft. 
.-.  138  yd.  2  ft.  of  carpet  are  needed. 

Make  a  diagram  for  this  question  on  the  scale  of  \  in.  to 
1ft. 

(3)  Reduce  6498  da.  to  wk. 

In  this  question  the  time  is  expressed  in  terms  of  the  primary  unit,  1  da., 
and  we  are  required  to  express  it  in  terms  of  the  derived  unit,  1  wk.  or  7  da. 
Dividing  6498  da.  by  7  da.,  the  result  is  928  wk.  2  da. 


Exercise  34 

In  the  following  exercise  prove  the  correctness  of  your  answers : 

1 .  Reduce  to  qt. :    36   pt. ;     78  pt. ;     96  pt. ;     65  pt. ;     and 
257  pts. 

2.  Find  the  number  of  strips  of  carpet  2  ft.  wide  required  to 
carpet  rooms  respectively  24,  28,  and  32  ft.  wide. 

3.  Reduce    to    yd.:     384   ft.;     456   ft.;     723   ft.;     897    ft.; 
5280  ft. 


DIVISION  63 

4.  Find  the  number  of  strips  of  carpet  3  ft.  wide  required  to 
carpet  rooms  respectively  27,  33,  and  39  ft.  wide. 

5.  Find  the  number  of  yards  of  carpet  required  to  carpet  a 
room  24  ft.  long  and  18  ft.  wide,  the  carpet  being  2  ft.  wide  and 
running  lengthwise.     Draw  a  diagram. 

6.  In  question  5,  find  the  number  of  yards  required  in  case 
the  carpet  runs  across  the  room,  and  draw  the  diagram. 

7.  Eeduce  to  gal.  :  576  qt. ;  893  qt. ;  798  qt. ;  962  qt. 

8.  Divide  by  5  :  475  ;  827  ;  593  ;  890  ;  646. 

9.  Divide  by  6:  252;  435;  728;  846;  777. 

10.  Eeduce   to   wk. :    245   da. ;    365   da. ;    678   da. ;   899   da. ; 
987  da. 

11.  Eeduce   to    pk :     32   qt. ;     892    qt. ;     958   qt. ;    2456    qt.  ; 
9472  qt. 

12.  Eeduce  to  sq.  yd.:  756  sq.  ft.;  894  sq.  ft.;  3478  sq.  ft.; 
9864  sq.  ft. 

13.  Eeduce    to    dimes:     620^;     840^;    729^;     843^;    5246^; 
8795^. 

14.  Divide  by  11:  451;  628;  847;  956;  8297;  7887. 

15.  How  many  dozen  are  there  in  842  units  ?  957  units  ?  1459 
units  ?  4596  units  ? 

16.  Eeduce  to  ft. :  459  in. ;  897  in. ;  2641  in. ;  63,360  in. 

17.  Find  the  length  of  the  sides  of  squares  whose  areas  are 
respectively  1,  4,  9,  16,  25,  36,  49,  64,  81,  100,  121,  and  144  sq.  in. 

18.  Find  the  length  of  an  oblong  6  ft.  wide  which  contains 
258  sq.  ft. 

19.  What  is  the  width  of  an  oblong  which  contains  15  units  of 
area,  the  length  containing  5  units  of  length,  and  the  unit  of 
length  being  4  ft.  ? 


64 


ARITHMETIC 


Exercise  35 


Find  the  quotient  and  remainder  and  prove  your  results  correct 

1.  36-2;  48-3;  72-4;  65-5;  84-6. 

2.  56-7;  96-8;  81-9;  80-10;  77-11;  48-12. 

3.  219-3;  842-6;  941-8;  654-9. 

4.  137-2;  439-5;  849-7;  999-12. 


5. 

6. 
7. 
8. 
9. 
10. 
11. 
12. 

6)743 

11)682 
4)673 
10)6430 

2)2557 
12)8256 
8)7919 

9)847 
8)976 
9)1978 
11)7381 

10)895 
12)899 
12)4994 
3)4565 
4)3191 
7)6769 

5)679 

7)2457 

6)3975 
6)4544 
11)8149 
4)2319 

5)1935 
9)4676 

5)9847 
8)7798 

6)1764 

12)9543 

11)8682 

9)7992 

12)63360 

Divide  by  short  division  : 
13.  200)14800    200)148 
14.  300)29654    400)987 

76    300)14976 

43    500)25931 

69.  (1)  A  father  dying  left  an  estate  valued  at  148,832  to 
be  divided  equally  among  his  wife,  his  two  sons,  and  his  four 
daughters.  What  was  the  share  of  each  ? 

In  this  problem  we  are  required  to  find  the  share  of  each,  which  is  the 
unit  of  measure.  In  order  to  find  this,  we  are  given  the  value  of  the  estate, 
which  is  the  whole  quantity,  and  one  factor,  which  is  the  number  of  shares  ; 
viz.  1  +  2  -f  4,  or  7.  Therefore,  dividing  the  whole  quantity  by  7,  we  find 
the  share  of  each  to  be  $  6976. 

(2)  The  area  of  the  four  walls  of  a  room  whose  dimensions 
are  8  yd.  and  6  yd.  is  112  sq.  yd.  Find  the  height  of  the 
room. 


DIVISION  65 

We  are  here  required  to  find  the  number  of  yd.  in  the  height  of  the 
roorn.  In  order  to  find  it,  we  are  given  the  area,  which  is  the  measured 
whole. 

We  are  also  given  the  dimensions  with  which  we  can  find  the  perimeter  of 
the  rooms,  and  thus  find  the  measuring  unit  as  in  §  54. 

The  perimeter  =  twice  the  sum  of  6  -f  8,  or  14  yd.  =  28  yd. 

The  area  of  a  strip  1  yd.  wide  running  around  the  room  =  28  sq.  yd. 

The  number  of  yd.  in  the  height  =  112  sq.  yd.  -f-  28  sq.  yd.  =  4. 

.  • .  the  height  =  4  yd. 

70.  If  we  multiply  59  by  724,  we  do  so  by  the  following 
process : 

59  59 

724  724 


236        or,  rearranging  the  work,        413 

118          we  have  this:  118 

413  236 


42716  42716 

We  now  wish  to  arrive  at  a  method  of  recovering  724 
from  the  dividend  42,716  and  the  divisor  59.  It  is  evident 
that  if  from  42,716  we  subtract  413  of  the  hundreds'  digit, 
which  is  the  product  of  59  and  7  times  the  hundreds'  digit, 
and  from  the  remainder,  118  of  the  tens'  digit,  which  is  the 
product  of  59  and  twice  the  tens'  digit,  and  from  this 
remainder,  236,  which  is  the  product  of  59  and  4,  we  shall 
have  no  remainder. 

• 

Hence  to  divide  42,716  by  59,  we  have  the  following 
method : 

First  divide  59  into  427  to  get  the  quotient  7  of  the 
hundreds'  digit,  multiply  57  by  7,  and  subtract  the  prod- 
uct 413  from  427,  leaving  14  of  the  hundreds'  digit,  which 
with  the  1  of  the  tens'  digit  makes  141  of  the  tens'  digit. 


66  ARITHMETIC 

Divide  this  141  by  59,  and  we  have  the  quotient  2  of  the 
tens'  digit.  Multiply  59  by  2,  and  subtract  the  product  from 
141,  leaving  23  of  the  tens'  digit,  which  with  the  6  makes 
236.  Divide  236  by  59,  and  we  have  the  quotient  4.  Multi- 
ply 59  by  4,  and  subtract  the  product  from  236,  and  there 
is  no  remainder. 

59)42716(724 
413 


141 

118 

236 
236 


Divisor  Dividend  Quotient 

71.   (1)  31  )  7598  (  245 

62 


139 
124 

158 
155 


3  Remainder 


31  divides  75  of  the  hundreds'  unit  two  hundred  times.  Put  2  as  the  first 
term  in  the  quotient. 

Multiply  31  by  2,  and  subtract  the  product  62  from  75.  The  remainder  is 
13  of  the  hundreds'  unit.  Annex  the  9  tens  of  the  dividend,  making  139 
tens.  31  divides  139  tens  4  tens  times.  Put  4  as  the  second  term  in  the 
quotient.  Multiply  31  by  4,  and  subtract  the  product  124  from  139.  The 
remainder  is  15  of  the  tens'  unit.  Annex  the  8  units,  making  158  units.  31 
divides  158  units  5  times.  Put  5  as  the  third  term  in  the  quotient.  Multiply 
31  by  5,  and  subtract  the  product  155  from  158.  The  remainder  is  3. 

What  is  the  quotient  on  dividing  7598  by  31  ?  What  does  it  show  ?  What 
is  the  remainder  ?  What  does  it  show  ?  See  §  63. 


DIVISION  67 

(2)  To   prove   that   the    answer   in    the   last  example  is 

correct : 

245  Quotient 

31  Divisor 

245 

735 


7595  Product 
3  Remainder 


7598  Dividend 


.-.  the  answer  is  correct.    Or  thus,  by  division, 

245)7598(31 
735 


248 
245 

3 

.-.  245  quotient  and  3  remainder  is  the  correct  answer. 

72.   Divide  39,726  by  87. 

87)39726(456 
348 


In  this  division  name  the  unit  to  which  each  remain- 
der and  each  partial  dividend  belongs. 


576 
522 

^4 

73.    Trial  divisor  and  trial  dividend. 

The  work  of  finding  the  quotients  can  be  much  simplified  by  using  the 
trial  divisor  and  trial  dividend. 

Thus  in  §  71,  as  31  is  nearer  30  than  40,  the  trial  divisor  is  3.  Dividing  3 
into  the  trial  dividends  7,  13,  and  15,  the  quotients  are  2,  4,  and  5. 

In  §  72,  as  87  is  nearer  90  than  80,  the  trial  divisor  is  9.  Dividing  9  into 
the  trial  dividends  39,  49,  and  57,  the  quotients  are  4,  5,  and  6. 


68 


ARITHMETIC 


In  general,  if  the  divisor  is  Gl,  02,  6:],  015,  027,  or  034,  the  trial  divisor 
is  6. 

If  the  divisor  is  07,  08,  09,  075,  089,  or  007,  the  trial  divisor  is  7. 

If  the  divisor  is  04,  05,  or  00,  the  use  of  the  trial  divisor  is  less  certain  ; 
but  the  rule  is  to  use  6  as  the  trial  divisor  for  04,  7  for  00,  and  either  0  or  7 
for  05. 

Name  the  trial  divisors  in  the  next  exercise. 


Exercise  36 


Find  the  quotients  and  remainders  of  the  following,  and 
prove  the  answers  to  the  odd  numbers  correct  by  multiplying, 
and  the  even  numbers  correct  by  dividing. 


1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 
10. 
11. 
12. 
37. 


712  --  31. 
2341  -  51. 

6287  -  71. 
2195  -  80. 
5894  -  91. 
2068  -  22. 
3572  -  42. 
1576  -  62. 
8189  -  82. 
6285  -  23. 
7549  -  53. 
8476  -  63. 


13.  9989-93. 

14.  7948-29. 

15.  8543-49. 

16.  9765-69. 

17.  8720-89. 

18.  8888-38. 

19.  9894-18. 

20.  9320-58. 

21.  16,324-78. 

22.  30,086-98. 

23.  18,874-27. 

24.  21,803-37. 


25.  36,989-67. 

26.  52,298-87. 

27.  75,643-97. 

28.  23,877-24. 

29.  38,753-34. 

30.  63,056-64. 

31.  74,111-25. 

32.  96,433-75. 

33.  56,159-95. 

34.  27,766-36. 

35.  56,139-56. 

36.  78,045  -  76. 


Reduce  9.872  qt,  to  bu.  (1  bu.  =  =  32  qt.). 


38.    Make  simple  practical  problems  based  on  questions  26-36. 

State  how  to  find  the  quotient  and  remainder  in  any  ques- 
tion in  long  division.  In  the  following  exercise,  before  divid- 
ing, make  a  careful  guess  as  to  what  the  quotient  will  be. 


DIVISION 


69 


1.  10,377-13. 

2.  29,452-14. 

3.  99,624-15. 

4.  87,643-16. 

5.  63,277-17. 

6.  64,935-18. 

7.  99,658-19. 

8.  29,943-99. 


Exercise  37 

9.  37,847-86. 

10.  84,374-45. 

11.  22,158-23. 

12.  84,999-69. 

13.  15,273-34. 

14.  42,965-88. 

15.  335,296-47. 


17.  854,300-49. 

18.  537,047-36. 

19.  624,839-75. 

20.  802,666-33. 

21.  263,204-54. 

22.  467,989-68. 

23.  467,989-67. 


25.  604,826-29. 

26.  253,789-96. 


16.    582,934-56.         24.    633,600-76. 

27.  494,358-65. 

28.  832,016-79. 


In  the  following  exercise,  before  dividing,  make  a  careful 
guess  as  to  what  the  quotient  will  be. 


1.  395,267-105. 

2.  300,498-207. 

3.  227,876-121. 

4.  407,253-309. 

5.  839,428-224. 

6.  719,888-421. 

7.  584,287-593. 

8.  495,638-784. 

9.  597,445-656. 

10.  386,777-921. 

11.  811,394-675. 


Exercise  38 

12.  $367,989-476. 

13.  578,243  cu.  in.  -  231  cu.  in. 

14.  578,243  rd.  -320  rd. 

15.  987,655  cu.  ft.  -  128  cu.  ft. 

16.  $128,821-360. 

17.  599,647-176. 

18.  313,947  da.  -365  da. 

19.  444,555-366. 

20.  574,381  A.  -  640  A. 

21.  987,432  sq.  in.  —  144  sq.  in. 


22.   358,049-528. 

23.  Solve  Exercise  22  by  division. 

24.  Make  simple  practical  problems  based  on  questions  12-22. 


70  ARITHMETIC 

Exercise  39 

1.  Divide  $  324  among  A,  B,  and  C,  giving  B  twice  as  much 
as  A,  and  C  three  times  as  much  as  B. 

2.  A  rod  540  in.  long  has  a  piece  of  8  in.  cut  off  from  it, 
then  another  piece  of  the  same  length,  then  another,  and  so  on. 
How  often  may  this  be  done,  and  what  is  the  length  of  the  piece 
remaining  at  last  ? 

3.  The  sum  of  $15,108  was  paid  for  a  number  of  sheep  at 
$  6  apiece.     Find  the  number  of  sheep. 

4.  What  is  the  object  of  division  when  the  measuring  unit 
and  the  measured  quantity  are  given  ?     When  the  number  and 
the  measured  quantity  are  given  ? 

5 .  A  merchant  sold  a  quantity  of  silk  at  $  3  a  yd.,  and  an 
equal  quantity  at  $  5  a  yd.      How   much  did  he   sell  of  each 
kind  if  he  received  $  3816  for  the  goods  ? 

6.  A  merchant  sold  a  quantity  of   cloth  at  $  3  a  yd.,  and 
twice   as  much  at   $  2  a  yd.,  the  whole  amounting  to  $  2065. 
How  much  did  he  sell  altogether? 

f 

7.  What  number  must  be  added  to  91  to  make  it  exactly 
divisible  by  8  ? 

8.  A  farmer  mixed  15  bu.  of  oats,  worth  40^  per  bu.,  with 
5  bu.  of  corn,  worth  80^  per  bu.     What  is  the  mixture  worth 
per  bu.  ? 

9.  The  expense  of  carpeting  a  room  was  $45;    but  if  the 
breadth  had  been  3  ft.  less  than  it  was,  the  expense  would  have 
been  $  36.     Find  the  breadth  of  the  room. 

10.  How  much  water  must  be  mixed  with  600  gal.  of  wine,  at 
$  2.50  per  gal.,  in  order  to   make  the  mixture  worth   $  2   per 
gal.  ? 

11.  Divide  $  448  among  2  men,  3  women,  and  4  children,  giv- 
ing each  man  three  times  and  each  woman  twice  as  much  as  each 
child. 


DIVISION  71 

Exercise  40 

1.  493,287  yd. -5- 1760  yd.  8.  819,634-4972. 

2.  298,456  ft. -s- 5280  ft.  9.  819,634-3264. 

3.  140,008  cu.  in. -j- 1728  cu.  in.  10.  205,639  -  7459. 

4.  680,442  cu.  in. -2150  cu.  in.  11.  726,998  -  9543. 

5.  998,209  Ib.- 2240  lb.  12.  337,877-9961. 

6.  857,864  gr.  -s-  5760  gr.  13.  698,206-8456. 

7.  398,125  gr. -s- 7000  gr.  14.  729,453-5879. 

Exercise  41 

1 .  Find  the  number  of  mi.  in  56,978  rd. 

2.  Find  the  number  of  mi.  in  86,300  yd. 

3.  Find  the  number  of  mi.  in  34,720  ft. 

4.  Find  the  number  of  mi.  in  746,360  in. 

5.  Find  the  number  of   sq.  ft.  in  a  room  263  in.  long  and 
248  in.  wide. 

6.  Find  the  number  of  townships,  each  containing  36  sq.  mi., 
which  can  be  made  out  of  a  section  of  land  in  the  form  of  an 
oblong  27  mi.  long  and  12  mi.  wide. 

7.  If  1  cu.  ft.  contains  1728  cu.  in.,  find  the  number  of  cu.  ft. 
in  632,194  cu.  in. 

8.  If  1  gal.  contains  231  cu.  in.,  find  the  number  of  cu.  ft. 
in  576  gal. 

9.  Find  the  largest  number  of  gal.   of  water  which  can  be 
emptied  into  a  vessel  containing  1  cu.  ft.,  without  overflowing. 

10.  How   many  gal.    of   water    will   fill    a   vessel    containing 
77  cu.  ft.  ? 

11.  Find  the  number  of  yr.,  of  365  da.  each,  in  84,678  da.. 

12.  Find  the  wages  due  a  workman  who  has  worked  423  hr. 
at  $  1.50  a  da.,  of  9  hr.  each. 


72  ARITHMETIC 

13.  Bought  640  bu.  of  barley  at  the  rate  of  32  bu.  for  $  20.08, 
and  sold  it  at  the  rate  of  10  bu.  for  $  8.75.     Find  my  profit  on 
the  transaction. 

14.  The   earth's  polar   diameter  contains   41,707,796  ft.,  and 
the  difference  between  the  equatorial  and  the  polar  is  one-292nd 
part  of  the  latter.     Find  the  difference  between  the  two  in  miles. 

15.  A  farmer  bought  land  from  B  at  $  60  per  A.,  and  the 
same  quantity  from  C  at  $  85  per  A.     The  whole  amounted  to 
$  53,215.     How  many  A.  did  he  buy  from  each  ? 

16.  What  number  must  be  added  to  7,869,456  to  render  it  ex- 
actly divisible  by  8975  ? 

17.  $  90.90  are  shared  among  4  men,  5  women,  and  6  children, 
so  as  to  give  to  each  man  twice  as  much  as  to  each  woman,  and  to 
each  woman  three  times  as  much  as  to  a  child.     What  do  the 
women  get  ? 

18.  A  farmer  laid  out  $  71,778  in  purchasing  an  equal  number 
of  sheep,  hogs,  and  cows.     Each  sheep  cost  $  6,  each  hog  twice  as 
much  as  a  sheep,  and  each  cow  twice  as  much  as  a  hog.     How 
many  of  each  did  he  buy  ? 

19.  A  speculator  gave  $  18,810  for  horses,  and  sold  a  certain 
number  of  them  for  $  7990,  at  $  85  each,  losing  thereby  $  10  each. 
For  how  much  each  must  he  sell  the  remainder  so  as  to  gain 
$  2180  on  the  whole  ? 

20.  In  161,384  in.,  how  many  mi.  ? 

21.  If   68  bales   of   linen  contain   67,048  yd.,  and  each  bale 
contains   34  pieces,   and    each  piece  the    same   number   of  yd., 
how  many  yd.  are  there  in  each  piece  ? 

22.  If  the  quotient  is  5000,  when  the  divisor  is  2001  and  the 
remainder  100,  what  is  the  dividend  ? 

23.  Divide  10,149  by  7,  and  the  quotient  by  5;  thence  deduce 
the  true  remainder,  and  show  that  it  is  the  same  as  after  the 
division  of  10,149  by  35. 


DIVISION  73 

24.  What  number  is  that  to  which  if  38  and  5  times  38  be 
added,  and  the  sum  so  found  be  increased  by  7  times  itself,  the 
total  sum  is  2400  ? 

25.  A  dealer  in  horses  gave  $  9900  for  a  certain  number,  and 
sold  a  part  of  them  for  $  3825  at  $  85  each,  and  by  so  doing  lost 
$  5  a  head.    For  how  much  per  head  must  he  sell  the  remainder 
so  as  to  gain  $  1140  on  the  whole  ? 

26.  A   receives    011   225    shares   of   mining    stock   an   annual 
dividend  of  $  96  a  share ;  and  B  receives  the  same  total  annual 
dividend  on  270  shares  of  oil  stock.    Find  the  annual  dividend  on 
one  share  of  B's  stock. 

27.  A  drover  bought  a  number  of  cattle  for  $  17,100  and  sold  a 
certain  number  of  them  for  $  12,474  at  $  126  a  head,  gaming  on 
those  he  sold  $2574.     How  many  did  he  buy  at  first,  and  how 
much  did  he  gain  on  each  sold  ? 

28.  The  fore  wheel  of  a  carriage  is  8  ft.  in  circumference,  and 
in  a  distance  of  13  mi.  makes  2340  revolutions  more  than  the 
hind  wheel.     Find  the  circumference  of  the  hind  wheel. 

29.  Divide  $2640.75  among  4  men,  6  women,  and  8  children, 
giving  to  each  child  double  a  woman's  share,  and  to  each  woman 
triple  a  man's  share. 

30.  A  grain  merchant  bought  40,640  Ib.  of  wheat  at  $1.20 
per  bu.,  and  shipped  it  to  New  York  at  an  expense  of  3^  per 
bu.     Before  he  sold  it  there  was  a  loss  in  handling,  etc.,  of  -^ 
of  the  original  weight ;  his  profit  on  the  transaction  was  $  69.85. 
At  what  price  did  he  sell  the  wheat  ?    (A  bu.  of  wheat  weighs 
60  Ib.) 

31.  How  many  sq.  ft.  of  glass  are  required  to  glaze  5  windows, 
each  containing  14  panes  of  glass,  the  panes  measuring  17  in.  by 
15  in.  ? 

32.  If  the  average  number  of  people  to  the  sq.  mi.  is  called 
the  unit  of  population,  find  to  the  nearest  integer  this  unit  for 
your  own  state  and  also  for  the  following  states : 


74  ARITHMETIC 


STATE  ME?1™  POPULATION  (1890) 

Massachusetts,  8,315  2,238,943 

New  York,  49,170  5,997,853 

Illinois,  56,650  3,826,351 

Texas,  265,780  2,235,523 

California,  158,360  1,208,130 

South  Dakota,  77,650  328,808 

Account  for  the  difference  in  the  units  of  population. 

33.  If  the  unit  of  population  for  one  state  is  16  times  as  great 
as  that  of  another,  but  the  area  of  the  second  is  4  times  as  great 
as  the  first,  compare  the  population  of  the  two  states. 

34.  The  length  of  the  Missouri-Mississippi  Eiver  is  4200  mi., 
and  the  area  drained  by  it  is  1,250,000  sq.  mi.     If  this  area  is 
conceived  of  as  a  rectangle  whose  length  is  4200  mi.,  find  its 
width  to  the  nearest  integer. 

35.  Find,  as  in  question  34,  the  width  of  the  rectangle  for 
these  rivers: 

p  LENGTH  IN  AREA  DRAINED 

MILES  IN  SQUARE  MILES 

St.  Lawrence,  2000  350,000 

Amazon,  4000  2,500,000 

La  Plata,  2300  1,250,000 

74.   Divide  77.968  by  8. 

8)  77.968          8  is  contained  in  77  units  9  times  with  remainder  5  units.     8  is 

g  74g     contained  in  59  tenths  7  tenths  times  with  remainder  3  tenths. 

8  is  contained  in  36  hundredths  4  hundredths  times  with  remain- 

der 4  hundredths.    8  is  contained  in  48  thousandths  G  thousandths  times  with 

no  remainder. 

Hence  the  operation  of  dividing  a  dividend  containing  a  decimal  is  similar 
to  that  of  dividing  when  the  dividend  does  not  contain  a  decimal.  Care 
must  be  taken  to  insert  the  decimal  point  in  the  quotient  as  in  the  example, 
immediately  after  the  units'  figure  is  used  in  the  dividend. 


DIVISION  75 

Exercise  42 

Divide : 

1.  12.6  by  6.  5.    596.36  by  17. 

2.  7.56  by  7.  6.   889.92  by  72. 

3.  16.38  by  13.  7.    195.2544  by  473. 

4.  239.76  by  37.  8.    192.4947  by  171. 
9.  Divide  389.904  A.  of  land  equally  among  16  persons. 

10.  If  43  bu.  of  wheat  cost  $  37.625,  find  the  cost  of  1  bu. 

11.  Divide  the  following  numbers  by  10 : 

47.39 ;  543.21 ;  62 ;  7.64. 

12.  Write  down  the  quotients  obtained  on  dividing  the  follow- 
ing numbers  by  10 : 

64.52;  3.9;  742.63;  95.614. 

13.  State  how  to  find   the   quotient  without   actual  division, 
when  a  decimal  is  divided  by  10. 

14.  Divide  by  100  : 

792.6  ;  8943.62  ;  54.15  ;  89467.1. 

15.  State  how  to  find  the  quotient  without   actual   division, 
when  a  decimal  is  divided  by  100.     By  1000. 


CHAPTER   VII 

COMPARISON  OP  NUMBERS 

75.    If  $4  be  multiplied  in  turn  by  the  numbers  1,  2,  3,  4, 

5,  6,  7,  8,  9, 10, 11, 12,  the  products  will  be  $4,  $8,  $12,  $16, 
$20,  $24,  $28,  $32,  $36,  $40,  $44,  $48. 

Hence,  taking  $4  as  the  unit  of  measure,  and  noting  how 
often  the  unit  is  repeated  to  produce  $4,  $12,  $8,  $20,  $36, 
we  find  that  these  quantities  are  represented  by  1,  3,  2,  5, 
and  9  units,  respectively. 

Exercise  43 

What  is  the  largest  unit  which  will  measure  each  of  the 
following  quantities,  and  what  is  the  number  of  units  in  each 
of  them  ? 

1.  $27,  $  24,  $  12,  $  30,  $  33,  $  36  (unit  $  3).  : 

2.  $  45,  $  10,  $  20,  $  55,  $  35,  $  60  (unit  $  5).  j 

3.  42,  18,  36,  6,  54,  and  60  sheep. 

4.  63,  84,  14,  35,  49,  and  77  bbl.  of  flour. 

5.  32,  16,  48,  56,  80,  and  96  min. 

6.  54,  27,  9,  81,  36,  and  99  men.  ' 

7.  80,  20,  90,  110,  70,  and  120  bu.  of  oats. 

8.  99,  66,  33,  55,  121,  and  132  T.  of  coal. 

9.  108,  24,  72,  132,  48,  and  60  Ib.  of  tea. 


76 


COMPARISON    OF   NUMBERS  77 

76.   If  25  sheep  cost  $120,  what  will  10  sheep  cost? 

Taking  5  sheep  as  the  unit  of  value,  25  sheep  contain  5  units,  and  10  sheep 

2  units. 

5  units  cost  $  120, 

1  unit  costs    $  24. 
.-.  2  units  cost    $  48. 

Exercise  44 

1.  If  36  yd.  of  cloth  cost  $42,  what  will  24  yd.  cost  at  the 
same  rate  ? 

2.  A  miller  sold  35  bbl.  of  flour  for  $126.     How  much  will 
he  receive  for  15  bbl.  at  the  same  rate  ? 

3.  A  train  runs  32  mi.  in  48min.     At  the  same  rate,  what 
distance  will  it  run  in  54  min.  ? 

4.  If  84  men  can  dig  a  trench  in  36  da.,  how  long  will  it 
take  108  men  to  dig  a  trench  of  the  same  size  ? 

5.  If  88  horses  eat  33  bu.  of  oats  in  1  da.,  how  many  bu. 
will  48  horses  eat  in  the  same  time  ? 

6.  If  45  men  can  reap  a  field  of  36  A.  in  a  certain  time,  how 
many  A.  would  25  men  reap  in  the  same  time  ? 

7.  A  bankrupt  pays  $  35  out  of  every  $  63  owed.    How  much 
shall  I  receive  if  I  am  his  debtor  to  the  extent  of  $  81  ? 

8.  If  32  T.  of  coal  cost  $  184,  what  will  88  T.  cost  at  the 
same  rate  ? 

9.  If  56  men  can  do  a  piece  of  work  in  21  da.,  how  long 
will  it  take  24  men  to  do  it  ? 

10.  How  many  Ib.  of  tea  can  be  bought  for  $56,  at  the  rate 
of  $  16  for  34  Ib.  ? 

11.  Tea  is  bought  at  72^  a  Ib.,  and  sold  for  84^  a  Ib.     The 
buying  price  is  what  part  of  the  selling  price  ? 

12.  The  cost  of  fencing  132  rd.  of  railway  is  $117.     What  is 
the  cost  of  fencing  88  rd.  ? 


78  ARITHMETIC 

13.  If  a  12-qt.  pail  is  just  filled  by  6  units  of  milk,  how  many 
qt.  are  there  in  a  pail  which  will  hold  5  units  ?     What  is  the 
unit  ? 

14.  Fifty-four  min.  are  represented  by  9  units  of  time.     How 
many  min.  are  there  in  7  units  ? 

15.  If  4  ft.  is  the  unit  of  length,  what  is  the  unit  of  area? 

16.  If  4  ft.  is  the  unit  of  length,  how  many  units  of  area  are 
there  in  an  oblong  whose  dimensions  are  24  and  36  ft.  ? 

17.  If  5  ft.  is  the  unit  of  length,  what  is  the  unit  of  volume  ? 

18.  If  5  ft.  is  the  unit  of  length,  how  many  units  of  volume 
are  there  in  a  rectangular  solid  30  ft.  long,  25  ft.  wide,  and  20  ft. 
high  ? 

19.  Divide  $84  among  two  persons,  giving  the  first  $3  for 
every  $  4  the  second  will  get. 

20.  Divide  $  88  between  A,  B,  and  C,  so  that  A  will  get  $  2 
and  B  $  4  for  every  $  5  C  gets. 

21.  A  sum  of  money  is  divided  between  A  and  B.     If  A,  who 
gets  $  2  for  every  $  5  B  gets,  receives  $  24,  how  much  does  B 
receive  ?     What  sum  was  divided  ? 

22.  A  sum  of  money  is  divided  between  A  and  B.     A  receives 
$  2  for  every  $  5  B  receives.     If  B  gets  $  24  more  than  A,  how 
much  did  each  receive  ?     What  sum  was  divided  ? 

23.  Make  and  solve  problems  similar  to  19,  20,  21.  and  22. 

24.  A  man  whose  weekly  income  is  $  15  spends  $  2  out  of 
every  $5  he  earns  for  board.     What  does   his  board  cost  him 
per  wk.  ? 

77.    (1)    What  is  the  ratio  of  1 20  to  1 45? 

Let  $5  be  taken  as  the  unit  of  measure. 
Then  $20  is  measured  by  4  times  the  unit. 
And  $45  is  measured  by  0  times  the  unit. 
.-.  $  20  is  f  of  $  45. 


COMPARISON  OF   NUMBERS  79 

(2)  If  22  yd.  of  cloth  cost  $16,  what  will  33  yd.  cost  at 
the  same  rate  ? 

Take  11  yd.  as  the  unit  of  length. 

Then  33  yd.  =  |  of  22  yd. 

/.  33  yd.  cost  »  of  $  16  or  $24. 

Exercise  45 

1.  What  is  the  ratio  of  $18  to  $  24?    $35  to  $55?     $28 
to  $  63  ? 

2.  What  is  the  ratio  of  16  hr.  to  56  hr.  ?     72  hr.  to  45  hr.  ? 

3.  What  is  the  ratio  of  60  mi.  to  25  mi.  ?     99  A.  to  55  A.  ? 

4.  If  45  cd.  of  wood  cost  $  162,  what  will  20  cd.  cost  ? 

5.  If  21  T.  of  hay  cost  $  174,  what  will  70  T.  cost  ? 

6.  If  63  men  can  dig  a  trench  in  16  da.,  how  long  will  it  take 
18  men  to  dig  it  ? 

7.  If  a  piece  of  cloth  15  ft.  long  and  3  ft.  wide  costs  $  18, 
what  will  a  similar  piece  20  ft.  long  and  4  ft.  wide  cost  ? 

8.  Divide  $  96  between  A  and  B  so  that  A  will  get  $  5  for 
every  $  7  B  will  get. 

9.  Divide  $  240  between  A  and  B  so  that  the  two  parts  will 
be  in  the  ratio  of  their  ages,  which  are  8  and  12  yr. 

10.  Divide  $  460  among  three  persons,  A,  B,  and  C,  so  that 
the  three  portions  will  be  to  each  other  as  the  numbers  5,  8,  and 
7,  respectively. 

11.  A  bankrupt  has  three  creditors,  to  whom  the  sums  due  are 
as  the  numbers  2,  3,  and  4.     If  his  assets  are  valued  at  $  540, 
find  the  sum  each  will  receive. 

12.  A  tract  of  land  is  divided  into  two  farms  in  the  ratio  of 
2  to  3.     If  the  whole  tract  contains  480  A.,  what  is  the  size  of 
each  farm  ? 


CHAPTER   VIII 
SQUAKE  ROOT 

78.  The  product  of  3  and  3  is  9 ;  of  5  and  5  is  25.     The 
squares  whose  sides  measure  3  and  5  units  of  length  contain 
9  and  25  units  of  square  measure.     We  say  that   9  is  the 
square  of  3  and  that  25  is  the  square  of  5 ;    that  3  is  the 
square  root  of  9  and  5  the  square  root  of  25. 

The  square  of  3  is  written  32,  and  the  square  root  of  9  is 
indicated  thus :  V9. 

2  is  called  the  Exponent,  and  ^J  the  Radical  Sign.  32  is  also 
called  the  second  Power  of  3. 

79.  The  Square  of  a  number  is  the  product  found  by  mul- 
tiplying the  number  by  itself. 

Thus  the  squares  of  1,  2,  3,  4,    5,    6,    7,    8,    9,    10, 
are  1,  4,  9,  16,  25,  36,  49,  64,  81,  100. 

80.  The  square  root  of  a  number  is  that  number  which 
multiplied  by  itself  is  equal  to  the  given  number. 

Thus  the  square  roots  of  1,  4,  9,  16,  25,  36,  49,  64,  81,  100, 

are  1,  2,  3,   4,    5,    6,    7,    8,    9,    10. 

81.  Pupils  should  memorize  the  tables  in  the  two  preceding  paragraphs 
and  be  able  to  answer  instantly  such  questions  as  the  following  : 

What  is  the  first  figure  in  the  square  root  of  27  ?   58  ?   76  ?   43  ?   80  ? 

Exercise  46 

Write  the  following  products  as  powers : 
1.    5  x  5.  2.    7  x  7.  3,   24  x  24. 

80 


SQUARE   ROOT  81 

Write  the  following  powers  as  products  and  find  their  values  : 

4.  82.  6.    462.  8.    (J)2. 

5.  132.  7.    (f)2.  9.    (ff)2. 

Prove  the  following  statements  : 


10.    V49  =  7.  13.    V324  =  18.  16. 


11.  V169  =  13.  14.    V961=31.  17.    VffH  =  fj. 

12.  V729  =  27.  15. 


Exercise  47 

1.  Find  the  squares  of  the  numbers  from  10  to  20  and  commit 
the  results  to  memory. 

2.  Find  the  squares  of  25,  28,  54,  75,  and  99. 

3.  From  the  results  in  §  79  state  how  many  digits  are  found 
in  the  square  of  a  number  of  1  digit. 

4.  From  the  results  obtained  in  questions  1  and  2,  state  how 
many  digits  are  found  in  the  square  of  a  number  of  2  digits. 

5.  Find  the  squares  of  175,  199,  246,  402,  814,  999. 

6.  From  the  results  in  question  5  state  how  many  digits  are 
found  in  the  square  of  a  number  of  3  digits. 

7.  Judging  from  the  results  obtained  in  questions  1,  2,  and  5, 
state  the  number  of  digits  in  the  square  root  of  a  square  number 
that  contains  3  digits  ;  4  digits  ;  6  digits  ;  7  digits  ;  8  digits. 

8.  How  many  digits  in  the  square  correspond  to  1  digit  in 
the  square  root  ? 

9.  What  is  the  square  root  of  400  ?     Of  900  ? 

10.  The  square  root  of  625  lies  between  what  two  numbers? 

11.  Find  the  square  of  10,  20,  30,  40,  50,  60,  70,  80,  and  90. 

12.  Between  what  numbers  does  the  square  root  of  1225  lie  ? 
Of  4225?     Of  2304?     Of  8281  ?     Of  8704  ? 

13.  What  is  the  square  of  200  ?     Of  300  ? 

G 


82  ARITHMETIC 

14.  The  square  root  of  71,289  lies  between  what  two  numbers  ? 

15.  Find  the  squares  of  100,  200,  300,  etc.,  up  to  900. 

16.  Between  what  two  numbers  does  the  square  root  of  271,441 
lie  ?     Of  795,664  ? 

82.    The  following  explanation  will  make  clear  the  method 
of  finding  the  square  root  of  a  number  of  3  or  4  digits. 

24  Thus  24,  which  is  made  up  of  two  parts,  20  and  4,  has  for  its 

<r>*       square  576,  which  is  seen  to  be  made  up  of  400,  the  square  of  20  ; 
—       16,  the  square  of  4  ;  and  twice  the  product  of  20  and  4. 

Now  to  recover  24  from  576,  we  know  that  its  hundreds'  digit  5, 

80      showing  that  the  number  is  between  400  and  900,  gives  the  tens'  digit  of 

80      the  root,  so  that  we  know  one  of  the  parts  of  the  root,  viz.  20.    The 

A(\(\      square  of  20  is  400,  and  the  rest  of  the  given  number,  176,  must  be 

— —       2  times  20,  multiplied  by  the  other  part,  together  with  the  square  of 

the  other  part.     Multiplying  20  by  2,  and  using  the  product  40  as  a 


20 


40 


576T20-I-4  divisor  with  176  as  dividend,  we  get  the  quotient  4. 
Multiplying  40  by  4  and  subtracting  the  product  160 
from  176,  we  get  the  remainder  16, 

5'76(24 


176  which  is  the  square  of  4.     Therefore 

160  20  +  4  or  24  is  the  square  root  of  576. 


-j  {*  The  work  of  extracting  the  square        \\ 


16 


root  may  be  simplified  by  leaving  out 
the  unnecessary  zeros,  thus : 


4 


176 
176 


83.  (1)  The  method  of  discovering  the  square  root  of  a 
number  of  5  or  6  digits  is  similar  to  that  for  finding  the 
square  root  of  numbers  of  3  or  4  digits. 

246 
246 

Thus  246,  which  is  made  up  of  two  parts,  240  and  6,  has  for 
1440  its  square  60,516,  which  is  seen  to  be  made  up  of  57,600,  the  square 
4440  of  240  ;  36,  the  square  of  6  ;  and  twice  the  product  of  240  and  6. 

57600 
60516 


44 


480 


6 


SQUARE  ROOT  83 

6'05'16(240  +  6  (2)  Hence  proceeding  with  605  as  in  §  82 

A  with  576,  we  get  in  the  square  root  24  tens  or 

240.    Multiplying  240  by  2  and  using  the  product 
480  as  a  divisor  with  2916  as  a  dividend,  the 


205 


176  quotient  is  found  to  be  6. 


291(5  Multiplying  the  480  2 

noon  by  6,  and  subtracting 

the  product  2880  from 

2916  we  have   the  re- 

36  mainder  36,    which  is 

the  square  of  6.    There-      486 


6'05'16(246 


205 
176 


2916 
2916 


fore  240  +  6  or  246  is  the  square  root  of  60516. 

Leaving  out  the  unnecessary  zeros,  the  work  may 
be  simplified  as  in  the  contracted  form. 

The  number  whose  square  root  is  to  be  subtracted  should  be  pointed  off 
into  groups  of  two  figures,  as  in  the  preceding  examples,  beginning  with  the 
units'  figure. 

84.  To  find  the  square  root  of  ||-|. 

The  square  roots  of  289  and  625  are  found  to  be  17  and  25. 

Hence  Vffjf  =  H- 

Exercise  48 
Find  the  square  root  and  prove  your  answer  correct : 

1.  324.  5.    3025.  9.    71,824.  13. 

2.  529.  6.   6889.  10.    101,761.  14. 

3.  841.  7.    4096.  11.    465,124.  15.    301 

4.  1156.  8.   9409.  12.   998,001.  16.   22^. 

17.  Compare  the  process  of  extracting  the  square  root  of  a 
number  with  long  division,  stating  in  what  respect  the  process  is 
similar,  and  where  it  is  different. 

85.  (1)   Find  the  length  of  the  side  of  a  square  containing 
4225  sq.  in. 

The  square  is  measured  by  4225  units  of  1  sq.  in.  ;  therefore  the  side 
is  measured  by  V4225,  or  65  units  of  1  in.,  i.e.  the  length  of  the  square 
is  65  in. 


64 

289' 


84  ARITHMETIC 

(2)  The  sides  of  a  rectangular  field  containing  735  sq.  rd. 
are  as  3  to  5.     Find  their  length. 

The  field  contains  3  x  5,  or  15  units  of  area. 

The  area  of  one  unit  =  735  sq.  rd.  -f-  15,  or  49  sq.  rd. 

The  side  of  a  square  containing  49  sq.  rd.  =7  rd. 

.-.  the  sides  of  the  field  are  3  x  7  rd.,  or  21  rd.,  and  5  x  7  rd.,  or  35  rd. 

STATEMENT  or  SOLUTION 

First  find  the  number  of  units  of  area  in  the  field.  Divide  this  number 
into  the  area,  and  find  the  unit  of  area.  Then  find  the  unit  of  length,  which 
is  the  length  of  the  side  of  the  unit  of  area. 

Multiply  the  unit  of  length  by  3  and  5  respectively,  to  find  the  sides  of  the 
field. 

(3)  To  find  the  area  of  a  triangle,  the  lengths  of  whose  sides  are  given, 
find  one-half  the  sum  of  the  number  of  units  of  length  in  the  sides.     Sub- 
tract from  this  the  number  of  units  of  length  in  each  side  separately.     Find 
the  product  of  these  four  results.     The  square  root  of  this  product  is  the 
number  of  units  of  area  in  the  given  triangle. 

Find  the  area  of  a  triangle  whose  sides  are  5  in.,  12  in., 
and  13  in.  respectively. 

The  sum  =  5  +  12  +  13  =  30. 
One-half  this  sum  =  15. 

15-     5  =  10.  15  x  10  x  3  x  2  =  900. 

15-12=    3.  V900  =  30. 

15  —  13  =    2.  .•.  the  area  of  the  triangle  =  30  sq.  in. 

Exercise  49 

1.  Find  the  length  of  the  side  of  an  enclosure  in  the  form  of 
a  square  containing  386  sq.  yd.  7  sq.  ft. 

2.  A  park  contains  9408  sq.  yd.,  and  it  is  3  times  as  long  as 
it  is  wide.     Find  its  length  and  width. 

3.  A  merchant  bought  a  number  of  yards  of  cloth,  paying  as 
many  cents  for  each  yard  as  there  were  yards.     The  whole  cost 
$  56.25.     How  many  yards  did  he  buy,  and  at  what  price  per 
yard? 


SQUARE   ROOT  85 

4.  What  is  one  of  the  two  equal  factors  of  24,336  ? 

5.  A  rectangular  field,  the  sides  of  which  are  in  the  ratio  of 
4  to  7,  contains  4032  sq.  rd.     Find  the  length  of  each  side,  and 
the  cost  of  fencing  it  at  $  4  per  rd. 

6.  A  body  of  soldiers  in  column  form  567  ranks,  7  abreast.    If 
they  were  drawn  up  in  solid  square,  how  many  would  there  be  on 
each  side  ? 

7.  Find  the  side  of  a  square  which  is  equal  in  area  to  the  sum 
of  the  area  of  two  squares,  the  sides  of  which  are  6  and  8  in. 
long. 

8.  Draw  two  lines   respectively  6  and   8  in.  long,  at  right 
angles.     Join  their  extremities  by  a  straight  line.     Measure  this 
line  and  show  that  it  is  equal  to  the  side  of  the  square  found  in 
question  7. 

9.  Work  problems  similar  to  7  and  8,  using  the  following  as 
the  lengths  of  the  sides  of  the  smaller  squares :  3  in.,  4  in. ;  5  in., 
12  in. ;  8  in.,  15  in. 

10.  From  the  preceding  three  questions  make  a  rule  showing- 
how  to  find  the  length  of  the  hypotenuse  of  a  right  triangle  when 
the  lengths  of  the  other  two  sides  are  known. 

11.  What  is  the  hypotenuse  of  a  right  triangle  whose  sides  are 
21  ft.  and  28  ft.  ?     15  ft.,  36  ft.  ?     56  ft.,  105  ft.  ? 

12.  Find  the  side  of  a  square  equal  in  area  to  the  difference  of 
the  area  of  the  two  squares  whose  sides  are  41  ft.  and  9  ft. 

13.  What  is  the  altitude  of  a  right  triangle  whose  hypotenuse 
and  base  are  34  ft.,  16  ft.  ?     205  ft.,  45  ft.  ?     136  ft.,  64  ft.  ? 

14.  The  top  of  a  ladder  rests  against  the  side  of  a  building  84 
ft.  from  the  ground,  and  its  foot  is  35  ft.  from  the  wall.     Find 
the  length  of  the  ladder. 

15.  A  ladder  51  ft.  long  stands  close  against  a  building.     How 
far  must  the  foot  be  drawn  out  that  the  top  may  be  lowered  6  ft.  ? 

16.  Find  the  diagonal  of  a  rectangular  field  whose  sides  are 
144  yd.  and  60  yd. 


86  ARITHMETIC 

17.  Find  the  side  of  a  square  equal  in  area  to  a  rectangle 
whose  sides  are  148  yd.  and  333  yd.     Find  the  difference  between 
the  perimeters  of  the  rectangle  and  square. 

18.  Find  the  area  of  the  largest  rectangle  which  can  be  en- 
closed by  a  line  36  in.  long. 

19.  A  field  in  the  form  of  a  rectangle  whose  sides  are  as  3  to  4 
contains  432  sq.  rd.     How  much  do  I  save  by  crossing  along  its 
diagonal  instead  of  going  along  its  two  sides  ? 

20.  State  in  as  few  words  as  possible  how  you  have  solved  each 
of  the  preceding  questions. 

21.  The  sides  of  a  triangle  are  8  in.,  15  in.,  and  17  in.     Find  its 
area. 

22.  Find  the  area  of  an  oblong  whose  sides  are  8  in.  and  15  in. 
What  is  the  ratio  of  its  area  to  that  of  the  triangle  given  in  ques- 
tion 21  ?     Why  is  this  so  ? 

23.  Find  the  areas  of  the  triangles  whose  sides  are  : 

21  in.,  28  in.,  35  in. 

24  in.,  45  in.,  51  in. 

9  in.,  40  in.,  41  in. 

24.  The  sides  of  a  rectangle  containing  34,992  sq.  ft.  are  as 
4  to  3.     Find  them. 

25.  One  side  of  an  oblong  is  f  as  long  as  the  other,  and  its  area 
is  67,335  sq.  yd.     Find  the  length  of  each  side. 


CHAPTER  IX 

GEEATEST  COMMON  MEASUEE  AND  LEAST  COMMON 

MULTIPLE 

86.  Name  all  the  units  of  length  which  will  exactly  meas- 
ure 15  in. 

They  are  1  in.,  3  in.,  5  in.,  and  15  in. 

87.  Find  all  the  different  units  of  length  that  will  exactly 
measure  12  ft.  and  18  ft. 

The  measures  of  12  ft.  are  1,  2,  3,  4,  6,  and  12  ft. 

The  measures  of  18  ft.  are  1,  2,  3,  6,  9,  and  18  ft. 

It  is  evident  that  all  the  common  measures  of  12  ft.  and 
18  ft.  are  1,  2,  3,  and  6  ft.,  and  that  the  greatest  common 
measure  is  6  ft. 

A  Common  Measure  of  two  or  more  quantities  is  a  unit  that 
will  exactly  divide  each  of  them. 

The  Greatest  Common  Measure  (G.  C.  M.)  of  two  or  more 
quantities  is  the  largest  unit  which  will  exactly  divide  each 
of  them. 

For  convenience  we  speak  of  the  common  measure  or  the 
greatest  common  measure  of  two  or  more  numbers. 

Exercise  50 

Find  all  the  common  measures  and  the  greatest  common  meas- 
ure of : 

1.  16ft,  28ft.  3.    54yd.,  72yd.  5.    32  qt.,  56  qt. 

2.  $  60,  $90.  4.    42  mi.,  105  mi.  6.    27  oz.,  47  oz. 

7.    21  pt.,  91  pt.  8.    84  bu.,  91  bu. 

87 


88  ARITHMETIC 

9.    Find  all  the  measures  that  can  be  used  to  measure  the 
capacity  of  each  of  two  baskets  containing  20  qt.  and  32  qt. 

10.  Find  the  lengths  of  the  two  longest  boards  that  can  be 
used  to  build  a  fence  around  a  garden  30  ft.  long  and  24  ft.  wide. 

11.  Make  questions  similar  to  9  and  10,  using  the  quantities 
in  problems  1  to  8. 

PRIME  NUMBERS 

88.  A  Prime  Number  is  one  that  can  be  divided  only  by 
unity  and  itself,  as  5,  11,  and  13. 

Select  the  prime  numbers :  2,  3,  4,  5,  6,  7,  8,  9,  10,  11. 

The  prime  factors  of  a  number  are  the  prime  numbers 
which  when  multiplied  together  give  it;  thus,  3,  3,  and  5 
are  the  prime  factors  of  45. 

89.  Find  the  prime  factors  of  168. 


2 
2 
2 
3 

168 

84 

42 

21 

7 

That  is,          168  =  2  x  84  ;   =  2  x  2  x  42  ;=  2  x  2  x  2  x  21 ; 

=  2x2x2x3x7; 
=  23  x  3  x  7. 

/.  The  prime  factors  of  168  are  2,  3,  and  7. 
23  is  a  short  way  of  writing  2x2x2. 

3  is  called  the  exponent  of  2,  and  denotes  that  2  has  been  taken  as  a  factor 
three  times. 

Exercise  51 

1.  Name  the  even  numbers  from  1  to  100. 

2.  Name  the  odd  numbers  from  1  to  100. 

3.  Name  the  prime  numbers  from  12  to  100. 
Find  the  prime  factors  of : 

4.  30;  36;  56;  48;  84;  66;  196;  195;  231. 


GREATEST   COMMON   MEASURE  89 

5.  86;  147;  104;  132;  78;  135;  342;  255. 

6.  336;  408;  372;  565;  342;  484;  375;  861. 

7.  What  prime  factors  are  common  to  30  and  36?     66  and 
132?     147   and  336?     135  and  255  ? 

90.  (1)  A  man  owns  a  rectangular  lot  210  ft.  long  and 
144  ft.  wide.  Find  the  length  of  the  longest  board  that 

can  be  used  to  fence  it. 

We  are  required  to  find  the  length  of  the  longest  board,  i.e.  the  G.  C.  M. 

of  144  ft.  and  210  ft. 

144  =  2x2x2x2x3x3. 

210  =  2  x  3  x  5  x  7. 

Thus,  the  G.  C.  M.  of  144  and  210  =  2  x  3  =  6. 
.-.  the  length  of  the  longest  board  is  6  ft. 
To  prove  the  answer  correct : 

The  number  of  boards  required  for  the  length  =  210  -=-  6  =  35. 

The  number  of  boards  required  for  the  width  =  144  -=-  6  =  24. 
35  and  24  have  no  common  measure  except  unity,  .-.  6  ft.  is  the  correct 
answer. 

(2)  A  certain  school  consists  of  132  pupils  in  the  high 
school,  154  in  the  grammar,  and  198  in  the  primary  grades. 
If  each  group  is  divided  into  sections  of  the  same  number 
containing  as  many  pupils  as  possible,  how  many  pupils  will 
there  be  in  each  section  ? 

We  are  required  to  find  the  number  of  pupils  in  each  section,  i.e.  the 
G.  C.  M.  of  132,  154,  and  198  pupils. 


2 
11 

132 

154 

198 

66 

77 

99 

6 

7 

9 

Since  2  and  11  are  the  only  common  factors,  the  G.  C.M.  of  132,  154,  and 

198  is  2  x  11,  or  22. 

.-.  each  section  will  contain  22  pupils. 

Exercise  52 

1.    Draw  two  lines,  one  15  in.  and  the  other  21  in.  long.     What 
is  the  longest  line  that  can  be  used  to  measure  both  lines  ? 


90  ARITHMETIC 

2.  What  is  the  longest  line  that  will  exactly  measure  two 
lines  28  and  32  in.  long  ? 

3.  What  is  the  longest  line  that  will  exactly  raeasnre  three 
lines  respectively  20,  30,  and  45  in.  long  ? 

4.  What  is  the  largest  nnit  of  capacity  that  can  be  used  to 
measure  the  quantity  of  oil  in  each  of  two  vessels,  one  containing 
16  qt.  and  the  other  36  qt.  ? 

5.  What  is  the  largest  unit  of  money  that  can  be  used  to  pay 
each  of  two  debts,  one  of  $  45  and  the  other  of  $  80  ? 

Exercise  53 

1.  A  certain  school  consists  of  132  junior  and  99  senior  stu- 
dents.    How  might  each  of  the  two  classes  be  divided  so  that  the 
whole  school  should  be  distributed  into  equal  sections  ? 

2.  A    gentleman   has   a   piece  of   ground,  the  sides  of  which 
measure  225  ft.,  297  ft.,  and  369  ft.     He  wishes  to  enclose  it 
with  a  fence   having   panels  of   uniform  length.     What  is   the 
longest  panel  that  can  be  used  for  that  purpose  ? 

3.  A  teacher  having  a  school  of  144  boys  and  128  girls  divided 
it  into  the  largest  possible  equal  classes,  so  that  each  class  of 
girls  should  number  the  same  as  each  class  of  boys.     What  was 
the  number  of  classes  ? 

4.  There  is  a  street  354  rd.  long,  and  the  land  on  one  side  of 
this  street  is  owned  by  three  persons,  A,  B,  and  C.     A  has  102 
rd.  fronting  the  street,  B  114  rd.,  and  C  138  rd.     They  agree  to 
divide  their  land  into  village  lots  in   such  a  manner  that  the 
lots  shall  be  of  the  greatest  width  that  will  allow  each  person  to 
form  an  exact  number  of  lots  out  of  his  land.     What  is  this 
width  ? 

5.  A  farmer  has  240  bu.  of  wheat  and  920  bu.  of  oats,  which 
he  desires  to  put  into  the  -least  number  of  boxes  of  the  same 
capacity,  without  mixing  the  two   kinds    of   grain.     Find   how 
many  bu.  each  box  must  hold. 


GREATEST   COMMON  MEASURE  91 

6.  If  1  lb.  Avoirdupois  contains  7000  gr.,  and  1  Ib.  Troy  5760 
gr.,  find  the  greatest  weight  that  will  measure  both  1  lb.  Troy  and 
1  lb.  Avoirdupois. 

7.  A  farmer  has  66  bu.  of  corn  and  90  bu.  of  wheat,  which  he 
wishes  to  put  into  sacks  of  equal  size,  and  without  mixing  the 
two  kinds  of  grain.     How  many  bu.  must  each  sack  contain  in 
order  to  be  as  large  as  possible  ? 

Exercise  54 

Find  the  G.  C.  M.  of : 

1.  40,  56.  9.  210,  455. 

2.  42,  54.  10.  287,  369. 

3.  81,  105.  11.  230,  506. 

4.  108,  162.  12.  42,  72,  180. 

5.  63,  91.  13.  60,  135,  165. 

6.  90,  105.                    .  14.  210, 462,  546. 

7.  102,  114.  15.  395,  474,  632. 

8.  75,  175.  16.  666,  738,  954. 

17.  Prove  that  your  answer  is  a  common  factor  by  dividing  it 
into  each  of  the  numbers.     Prove  that  it  is  the  greatest  common 
measure  by  examining  your  quotients  and  finding  that  they -have 
no  common  measure  except  unity. 

18.  State  how  to  find  the  G.  C.  M.  of  two  numbers.     Of  three 
numbers. 

19.  Make  questions  similar  to  those  in  the  previous  exercise. 

91.  Seven  divides  126  and  35.  It  also  divides  their  sum,  161,  23  times,  and 
their  difference,  91, 13  times.  Seven  also  divides  the  sum  of  35  and  4  x  126,  or 
539,  77  times,  and  the  difference  between  126  and  3  x  35,  or  21,  3  times. 

Thus  any  number,  as  7,  that  divides  two  other  numbers,  as  126  and  35, 
will  divide  their  sum  or  difference.  It  will  also  divide  the  sum  or  difference 
of  any  multiples  of  these  numbers. 


92  ARITHMETIC 

Exercise  55 

Prove  the  principle  stated  above  : 

1.  Divisor    6;     numbers    84,  30. 

2.  Divisor    8,     numbers    88,  24. 

3.  Divisor  13,     numbers    65,  26. 

4.  Divisor  19,     numbers  133,  38. 

92.  When  the  factors  of  the  number  cannot  be  easily  found,  the  follow- 
ing method  is  used : 

741)893(1 

741 

162)741(4 
608 

133)152(1 
133 
19)133(7 

133 

.'.  the  G.  C.  M.  of  741  and  893  is  19. 
To  prove  that  19  is  the  G.  C.  M.  of  741  and  893  : 

First.  19  divides  133,  and  therefore  it  divides  19  +  133,  or  152.  19  divides 
133  and  152,  and  therefore  it  divides  133  and  4  times  152,  or  741.  19  divides 
152  and  741,  and  therefore  it  divides  152  +  741,  or  893.  Therefore  19  is  a 
common  measure  of  741  and  893. 

Again.  It  is  also  the  G.  C.  M.  Any  number  which  divides  893  and  741 
will  divide  their  difference,  or  152.  Any  number  which  divides  741  and  152 
will  divide  the  difference  between  741  and  4  times  152,  or  133.  Any  num- 
ber which  divides  152  and  133  will  divide  their  difference,  or  19.  But  19  is 
the  largest  number  that  divides  19  ;  therefore  19  is  the  G.  C.  M.  of  741  and 
893. 

93.  The  following  more  compact  form  may  be  used  after  the  pupils  under- 
stand the  method  given  above.     It  will  be  observed  that  the  quotients  are 
omitted  and  that  the  divisors  741,  152,  133,  and  19  are  alternately  on  the 
left  and  right  side  of  the  vertical  line. 


741 

608 

893 
741 

133 
133 

152 
133 

19  G.  C.  M. 

GREATEST   COMMON  MEASURE  93 

94.  To  find  the  G.  C.  M.  of  three  numbers,  find  first  the 
G.  C.  M.  of  two  of  them,  and  then  of  the  result  and  the  third 
number. 

Exercise  56 

Find  the  G.  C.  M.  of: 

1.  145,  203.  7.  11,682,  19,626. 

2.  344,  559.  8.  31,416,  54,593. 

3.  465,  682.  9.  2487,  8413. 

4.  1781,  4384.  10.  495,  891,  1155. 

5.  3423,  3248.  11.  3066,  4818,  8541. 

6.  4807,  9545.  12.  15,561,  11,115,  13,585. 

13.    State  how  to  find  the  G.  C.  M.  of  two  numbers. 

Exercise  57 

1.  What  is  meant  by  saying  that  one  number  is  a  common 
measure  of  two  or  more  numbers  ?     Also,  the  greatest  common 
measure  f 

2.  Show  by  means  of  the  examples  in  the  preceding  exercise, 
that  the  greatest  common  measure  of  two  numbers   can  never 
exceed  the  difference  of  the  numbers. 

3.  In  the  following  pairs   of  numbers,  select  those  that  are 
prime  to  each  other,  and  those  that  are  not  prime  to  each  other : 
12,  18  ;  8,  15 ;  12,  17  ;  20,  21 ;  28,  35  ;  13,  29 ;  36,  48  ;  31,  47. 

4.  Explain  what  is   meant   by  one   number   being  prime   to 
another.     When  two  numbers  are  prime  to  each  other  are  they 
necessarily  prime  ?     Give  examples. 

5.  What  is  the  product  of  the  three  consecutive  numbers,  11, 
12,  13  ? 

6.  The  product  of  three  consecutive  numbers  is   120.     Find 
the  numbers. 


94  ARITHMETIC 

7.  The  product  of  four  consecutive  numbers  is  1680.     Find 
them. 

8.  Find  the  G.  C.  M.  of  90,  150,  and  168  by  resolving  the 
numbers  into  their  prime  factors.     When  several  numbers  have 
been  resolved  into  their  prime  factors,  which   of  these  factors 
must  be  taken  to  form  by  their  product  the   greatest  common 
measure  of  the  numbers  ? 

9.  Find  all  the  common  measures  of  210  and  462.     Prove 
that  every  common  measure  of  these  two  numbers  is  a  measure 
of  their  difference. 

10.  Find  the  G.  C.  M.  of  13,515  and  13,787. 

11.  How  many  rails  will  enclose  a  field  3143  ft.  long  by  2471 
ft.  wide,  provided  the  fence  is  straight  and  8  rails  high,  and  the 
longest  that  can  be  used  ? 

12.  Find  the  G.  C.  M.  of  169,037  and  66,429. 

LEAST  COMMON  MULTIPLE 

95.  The   quantity  15   in.  is  measured  by  the  unit  5  in., 
3  times,  and  is  therefore  called  a  multiple  of  5  in. 

The  quantity  18  Ib.  is  exactly  divisible  by  the  units  1  lb., 
2  lb.,  3  lb.,  6  lb.,  and  9  lb.,  and  is  a  multiple  of  each  one  of 
them.  Thus  18  lb.  is  equal  to  18(1  lb.),  9(2  lb.),  6(3  lb.), 
or3(61b.). 

Select  from  the  following  quantities  the  multiples  of  the 
unit  83:  112,  f  16,  $18,  $25,  and  827.  Name  all  the  units 
that  will  exactly  measure  the  quantity  24  hr. 

Any  quantity  is  a  multiple  of  a  unit  of  measure  when  it  is 
exactly  divisible  by  the  unit. 

96.  Thirty  clays  is  exactly  divisible  by  the  units  3  da.  and 
5  da.,  and  is,  therefore,  a  common  multiple  of  3  da.  and  5  da, 


LEAST   COMMON   MULTIPLE  95 

One  quantity  is  a  common  multiple  of  two  or  more  units 
when  the  former  is  exactly  divisible  by  each  of  the  latter. 

Thirty  days  is  the  least  quantity  that  is  exactly  divisible 
by  the  units  6  da.  and  10  da.,  and  is,  therefore,  the  least  com- 
mon multiple  of  6  da.  and  10  da. 

The  Least  Common  Multiple  (L.  C.  M.)  of  two  or  more  units 
is  the  least  quantity  that  is  exactly  divisible  by  each  of  them. 

For  convenience  we  speak  of  one  number  being  a  multiple 
of  another,  or  a  common  multiple,  or  the  least  common  mul- 
tiple of  two  or  more  numbers. 

97.  In  problems  in   Greatest   Common    Measure,   we  are 
given  two  or  more  quantities  and  are  required  to  find  the 
largest  unit  that  will  measure  each  of  them.     In   problems 
in  Least  Common  Multiple,  we  are  given  two  or  more  units 
of  measure  and  are  required  to  find  the  least  quantity  that 
can  be  measured  by  each  of  the  units. 

98.  3,  6,  9,  12,  15,  18,  21,  24,  are  multiples  of  3. 

2,  4,  6,  8,  10,  12,  14,  16,  18,  20,  22,  24,  are  multiples  of  2. 

.-.  6,  12,  18,  24,  are  common  multiples  of  2  and  3,  and  it  is 
evident  that  6  is  the  L.  C.  M.  of  2  and  3. 

The  second  common  multiple,  12,  is  2x6. 

The  third  common  multiple,  18,  is  3  x  6. 

The  fourth  common  multiple,  24,  is  4  x  6. 

What  are  the  fifth  and  sixth  common  multiples  of  2  and  3? 
In  the  same  way  find  common  multiples  and  the  L.  C.  M.  of 
2  and  5,  3  and  4,  4  and  6,  8  and  12. 

99.  The  preceding  paragraph  shows  that  the  L.  C.  M.  of  two  prime  num- 
bers, as  2  and  3,  2  and  5,  3  and  4,  are  their  products  6,  10,  12,  while  the 
L.  C.  M.  of  two  numbers  not  prime,  as  4  and  6,  and  8  and  12,  which  are  12 
and  24,  are  less  than  their  products,  and  are  in  both  cases  equal  to  the  prod- 
uct of  the  numbers  divided  by  their  G.  C.  M. 


96  ARITHMETIC 

100.    (1)  Find  the  shortest  distance  which  can  be  exactly 
measured  by  two  lines  respectively  36  ft.  and  48  ft.  long. 

We  are  here  required  to  find  the  shortest  distance,  i.e.  the  L.  C.  M.,  of  the 
units  36  ft.  and  48  ft. 

36  =  2x2x3x3. 
48  =  2x2x2x2x3. 

Thus  the  L.  C.  M.  of  36  and  48  =  2  x  2  x  2  x  2  x  3  x  3  =  144. 

.-.  the  shortest  distance  is  144  ft. 

(2)  Find  the  L.  C.  M.  of  24,  30,  36. 


6 


24     30     36 


6 


253 

Here  6  and  2  are  the  factors  common  to  two  or  more  of  the  numbers,  and 
2,  5,  and  3  are  the  factors  not  common. 

/.  the  L.  C.  M.  =  6  x  2  x  2  x  5  x  3  =  360. 

State  how  to  find  the  L.  C.  M.  of  two  or  more  numbers. 

Show  by  division  that  24,  30,  and  36  are  all  factors  of  their  L.  C.  M.  360. 

(3)  What  is  the  least  number  of  bu.  of  wheat  that  will 
make  an  exact  number  of  full  loads  for  three  drays,  hauling 
respectively  24,  30,  and  36  bu.  a  load  ? 

"We  are  required  to  find  the  least  number  of  bu.,  i.e.  the  L.  C.  M.  of  the 
units  24,  30,  and  36  bu.,  which  is  360  bu. 

To  prove  the  answer  correct  : 

Dividing  360  bu.  by  24,  30,  and  36  bu.  respectively,  we  find  the  number 
of  loads  to  be  15,  12,  and  10.  15,  12,  and  10  have  no  common  factor. 

/.  360  bu.  is  the  least  number  of  bu. 

101.    (1)  Find  the  L.  C.  M.  of  14,  21,  54,  56,  84. 


3 

2 


54     56     84 


18     56 


9    28 

V.  the  L.  C.  M.  =  3  x  2  x  9  x  28  =  1512. 

14  is  erased  since  it  is  a  factor  of  56,  and  21  since  it  is  a  factor  of  84.     In 
the  second  line,  28  is  erased  since  it  is  a  factor  of  56. 


LEAST   COMMON   MULTIPLE  97 


(2)  Find  the  L.  C.  M.  of  481  and  1665. 


481)1665(3 
1443 


222)481(2 
444 


37)222(6 
222 


481  -4-  37  =  13  ;  1665  -  37  =  45. 

.-.  the  L.  C.  M.  of  481  and  1665  =  37  x  13  x  45  =  481  x  45  =  21,645. 
State  how  to  find  the  L.  C.  M.  of  two  numbers. 
Show  by  division  that  481  and  1665  are  both  factors  of  21,645. 
Show  that  the  L.C.  M.  21,645  is  equal  to  the  product  of  481  and  1665 
divided  by  their  G.  C.  M.  37. 

Exercise  58 

Solve  the  following  problems  and  name  the  units  of  measure- 
ment: 

1.  How  long  is  the  L.  C.  M.  of  two  lines,  one  6  in.  long,  and 
the  other  10  in.  long?     How  many  times  is  it  measured  by  the 
6-in.  line  ?     By  the  10-in.  line  ? 

2.  Find  the  L.  C.  M.  of  three  lines  respectively  12  in.,  15  in., 
and  18  in. 

3.  Four  bells  toll  at  intervals  of  3,  7,  12,  and  14  sec.  respec- 
tively, and  begin  to  toll  at  the  same  instant.     When  will  they 
next  toll  together  ? 

4.  If  in  two  days  A  can  build  28  rd.  of   fencing,   B  50  rd., 
C  16  rd.,  and  D  40  rd.,  find  the  least  number  of  rd.  that  will 
furnish  an  exact  number  of  days'  work. 

5.  When  is  a  number   a  common  multiple  of   two  or  more 
numbers,  and  when  the  least  common  multiple  ? 

6.  What  is  the  least  number  of  A.  that  will  admit  of  being 
divided  into  a  number  of  farms  containing  150,  200,  or  250  A. 
each  ? 

7.  In  each  question,  prove  your  answer  correct. 

H 


98  ARITHMETIC 

Exercise  59 

Find  the  L.  C.  M.  of  : 

1.  4,  8,  16,  32.  4.    15,  18,  28,  36.  7.    65,  26,  56,  52. 

2.  3,  6,  9,  12.  5.    12,  20,  21,  45.  8.   36,  48,  60,  54. 

3.  24,  30,  36,  45.          6.   22,  33,  30,  44.  9.    33,  27,  55,  135. 
10.   30,  21,  40,  28,  24,  56.                11.  56,  36,  63,  28,  72. 

Find  the  L.  C.  M.  of  the  denominators  of  these  fractions  : 


15        1     5.     11     _8  -17        1        8         5        13 

5>    7>    21?    1^'  Ltm      3")  T5"?  TT 


1  fi        4      2.     1     1  4  1  Q        1      2        7        1  1 

5?    3?    9?    15'  10'      "3>   3"J   TY>  T8"' 

Exercise  60 
Find  the  L.  C.  M.  of  : 

1.  266,  703.  3.    3045,  4515.  5.    8159,  14,227. 

2.  1173,  1702.  4.   3589,  2257.  6.    10,959,  12,753. 

Exercise  61 

1.  Show  by  examples  that  the  L.  C.  M.  of  two  numbers  can 
never  exceed  their  product. 

2.  Find  the  L.  C.  M.  of  90,  150,  and  168  by  resolving  the  num- 
bers into  their  prime  factors.     When  several  numbers  have  been 
resolved  into  their  prime  factors,  which  of  these  factors  must  be 
taken  to  form  by  their  product  the  L.  C.  M.  of  the  numbers  ? 

3.  A,  B,  C,  and  D  start  together,  and  travel    the  same  way 
around  an   island  which  is   600  mi.  in  circuit.      A  goes  20  mi. 
per  da.,  B  30,  C  25,  and  D  40.     How  long  must  their  journeying 
continue,  in  order  that  they  may  all  come  together  again  ? 

4.  A  shepherd,  on  telling  his  sheep,  found  that  when  he  told 
them  out  by  twos,  threes,  fours,  and  fives,  he  had  none  left,  and 
he  knew  his  flock  was  above  300,  but  less  than  400.     What  num- 
ber had  he  ? 


LEAST   COMMON   MULTIPLE  99 

5.  Three  bodies  move  uniformly  in  similar  orbits  round  the 
same  centre  in  87,  224,  365  da.  respectively.     Supposing  all  three 
in  conjunction  at  a  given  time,  find  after  how  many  days  they 
will  be  in  conjunction  again. 

6.  An  island  is  48  mi.  in  circumference.     A,  B,  and  C  have 
to  walk  round  till  they  all  arrive  together  at  the  starting-point. 
A  walks  2,  B  3,  and  0  4  mi.  an  hour.     How  many  times  must 
each  go  round  before  the  task  is  accomplished  ? 

7.  Find  the  L.  C.  M.  of  11,  14,  28,  22,  7,  56,  27,  81,  54,  and  36. 

8.  If  1  Ib.  Avoirdupois  contains  7000   gr.,  and  1  Ib.  Troy 
5760  gr.,  find  the  least  weight  which  can  be  expressed  without 
fractions  in  both  Ib.  Troy  and  Ib.  Avoirdupois. 

9.  Prove  by  example  that  the  product  of  the  H.  C.  F.  and  the 
L.  C.  M.  of  two  numbers  is  equal  to  the  product  of  the  numbers, 
and  state  why  this  is  so. 

10.  The  L.  C.  M.  of  391  and  another  number  is  12,121,  and  the 
G.  C.  M.  is  23.     Find  the  other  number. 

11.  Along  a  certain  path  1600  yd.  long,  there  is  a  house  every 
50  yd.,  and  a  tree  every  20  yd.     How  many  houses  will  have 
a  tree  in  front  ? 

12.  The  periods  of  three  planets  which  move  uniformly  in  circu- 
lar orbits  round  the  sun,  are  respectively  200,  250,  and  300  da. 
Supposing  their  positions  relatively  to  each  other  and  the  sun  to 
be  given  at  any  moment,  determine  how  many  da.  must  elapse 
before  they  again  have  exactly  the  same  relative  positions. 


CHAPTER   X 

FKACTIONS 

102.  The  expression  5  ft.  denotes  a  quantity  measured  by 
the  number  5  and  the  unit  1  ft.     The  expression  5x2  in. 
denotes  a  quantity  measured  by  the  number  5  and  the  unit 
2  in.    The  expression  5  x  J  ft.  denotes  a  quantity  measured 
by  the  number  5  and  the  unit  one-sixth  of  a  ft. 

103.  The  expression  J  ft.  denotes  that  the  quantity  1  ft. 
is  conceived  as  made  up  of  6  equal  parts  or  units,  and  that 
5  of  these  parts  or  units  have  been  taken  to  measure  the 
quantity  denoted  by  |  ft. 

The  primary  unit,  1  ft.,  has  been  divided  into  6  equal 
parts  to  give  the  direct  measuring  unit,  which  is  ^  ft.,  or 
2  in.  The  number  of  these  units  in  the  given  quantity  is  5. 
The  ratio  of  the  given  quantity  to  the  direct  measuring  unit 
is  5. 

104.  The  quantity  represented  by  f  ft.  contains  5  direct 
measuring  units,  and  the  primary  unit,  1  ft.,  contains  6  of 
these  units.     Hence  the  fraction  ^  expresses  the  ratio  of  the 
quantity  denoted  by  J  ft.  to  the  primary  unit,  1  ft. 

C\ ! ! I ! \D 

A  I ! I !_         L_     I I  B 

105.  Draw  a  line,  AB,  1  ft.  long.     Divide  it  into  6  equal  parts,  or  units, 
each  i  of  a  ft.  long.     Draw  a  second  line,  CD,  above  the  first,  containing  5  of 
these  units,  and  use  these  two  lines  to  illustrate  the  preceding  paragraph. 

100 


FRACTIONS  101 

Exercise  62 

In  the  following  questions  give  the  number  of  direct  measuring 
units  that  makes  up  the  primary  unit.  Name  the  direct  measur- 
ing unit  in  two  ways.  Give  the  ratio  of  the  quantity  to  the  direct 
measuring  unit.  Give  the  ratio  of  the  quantity  to  the  primary 
unit. 

1.  f  ft.  5.    $f.  9.    -fj  sq.  ft.  13.    fda. 

2.  |  yd.  6.    $f.  10.    feu.  yd.  14.    f  da. 

3.  JJb,  7.    $TV         11.    fwk.  15.    ||  hr. 

4.  fsq.  yd.          8.    $J£.         12.    ff  yr.  16.    f  min. 


106.  A  Fraction  is  a  number  in  which  the  unit  of  measure 
is  a  definite  part  of  some  primary  unit  of  the  same  kind. 

The  denominator  shows  into  how  many  parts  the  primary 
unit  is  divided  to  give  the  direct  unit  of  measure  ;  it  also 
names  this  unit.  The  numerator  shows  the  number  of  them 
that  measures  the  quantity. 

A  proper  fraction,  as  an  expression  of  measured  quantity, 
is  one  in  which  the  numerator  is  less  than  the  denominator. 
Select  the  proper  fractions  :  -f,  -f-,  -|,  f  ,  f  . 

o 

A\  _  I  _  I  _  I  _  1_      _  \B 


D\ | | | 1 I 1  E 

107.  Let  AB  represent  some  quantity  measured  by  5 
units,  each  equal  to  AC,  and  DE  as  measured  by  6  units, 
each  equal  to  A  C.  Then  if  we  think  of  AB  in  relation  to 
DE,  we  think  of  5  units  in  relation  to  6  units,  and  this  rela- 
tion or  ratio  is  expressed  by  the  fraction  |. 

.Similarly,  the  fraction,  or  number  |,  expresses  the  ratio  of 
85  to  16,  5  hr.  to  6  hi\,  5  mi.  to  6  mi.,  5  (8  ft.)  to  6  (8  ft.), 
5  (12  Ib.)  to  6  (12  lb.),  or,  generally,  5  of  any  unit  to  6  of 
the  same  unit. 


102 


ARITHMETIC 


Similarly,  it  expresses  the  ratio  of  20  Ib.  (i.e.  5x4  Ib.) 
to  24  Ib.  (i.e.  6x4  Ib.),  and  so  on. 

Exercise  63 

Write  the  fraction  which  expresses  the  ratio  of  the  following 
quantities : 

1.  2  ft.  to  5  ft.;  2  ft.  to  1  yd. 

2.  $  4  to  $  9  ;  $  8  to  $  12  ;  1  dime  to  $  1. 

3.  3  qt.  to  4  qt. ;  2  qt.  to  1  gal. 

4.  12  yd.  to  32  yd. ;  4  in.  to  2  ft. 

5.  If  32  yd.  of  cloth  cost  $48,  what  part  of  $48  will  12  yd. 
cost  ?     How  many  dollars  will  12  yd.  cost  ? 

6.  24  men  to  36  men;  18  men  to  24  men. 

7.  If  18  men  can  do  a  piece  of  work  in  32  da.,  in  what  part 
of  32  da.  can  24  men  do  the  same  work  ?    How  many  da.  ? 

108.  Let  AB  represent  some  definitely  measured  quantity, 
as  4  ft.  or  16  ft.,  and  let  it  be  divided  as  shown  in  the 
diagram. 


FRACTIONS  103 

It  is  evident  from  this  diagram  that  f,  f,  |,  f ,  f,  f,  ^,  ^f, 
l|,  of  a  quantity  measure  it,  and  are  all  equal.  It  is  also  evi- 
dent that  ^,  J,  f ,  f ,  T5Q,  T62,  T8g,  of  a  quantity  measure  one-half 
of  it,  and  are  all  equal. 

Similarly,  ^,  f ,  y4^,  of  a  quantity  measure  one-third  of  it, 
and  are  equal.  Similarly,  ^  =  T2^. 

Exercise  64 

1.  Find  1  of  $  8 ;  f  of  $  8  ;  f  of  $  8. 

2.  Find  1  of  $30;  T%  of  $30;  fV  of  $30. 

3.  Find  1  of  36  ft. ;  find  also  respectively  f ,  -f,  T\,  ^-,  and  if 
of  36  ft. 

4.  Find  |  of  24  dimes  ;  f  of  24  dimes  ;  -^  of  24  dimes. 

5.  Find  f  of  28  Ib. ;  |J  of  28  Ib. ;  ff  of  28  Ib. 

6.  Find  |  of  32  da. ;  f  of  32  da. ;  if  of  32  da. ;  f  f  of  32  da. 

7.  Name  three  fractions  equal  to  -J;  three  equal  to  J--  three 
equal  to  f . 

8.  Name  two  fractions  each  equal  to  £,  and  prove  your  results 
correct  by  taking  each  fraction  of  $  24. 

Exercise  65 

1.  If  1  da.  is  made  up  of  24  measuring  units,  find  the  number 
of  such  units  respectively  in  -J-,  1,  1,  1,  ^  -JT,  of  a  da. 

2.  If  1  da.  is  made  up  of  24  measuring  units,  find  the  number 
of  such  units  respectively  in  i,  |,  |,  |,  ^  and  |J  of  a  da. 

3.  If  1  hr.  is  made  up  of  60  measuring  units,  find  the  number 
of  units  in  1,  -f^,  ^  of  an  hr. 

4.  If  1  wk.  contains   168  units  of  time,  how  many  units  of 
time  are  there  respectively  in  ^  T2¥,  ^-,  and  ^  of  a  wk.  ? 


104  ARITHMETIC 

5.  If  1  sq.  ft.  contains  144  units  of  area,  find  the  number  of 
units  of  area  in  f ,  -f ,  ^,  if,  and  ||  of  a  sq.  ft.  respectively. 

6.  If  $  1  contains  100  measuring  units,  find  the  number  of  units 
that  measure  $  f ,  $  T8o,  $  |-f ,  and  $  |J  respectively. 

7.  If  1  Ib.  is  represented  by  a  16-unit  quantity,  what  part  of 
a  Ib.  is  a  1-unit  quantity  ?  a  2-unit  quantity  ?  a  4-unit  quantity  ? 
an  8-unit   quantity  ?     What    part   of   a   pound    is   eight    1-unit 
quantities  ?   four  2-unit  quantities  ?  two  4-uiiit  quantities  ?  one 
8-unit  quantity  ? 

8.  What  actual  coins  and  how  many  of  each  kind  are  equal 
respectively  to  1,  J,  T%,  and  i{}  of  $  1  ? 

9.  From  the  above  examples  state  what  changes  can  be  made 
in  the  numerator  and  denominator  of  a  fraction  without  altering 
the  value  of  the  fraction. 

109.  In  this  exercise,  the  quantities  1  da.,  1  hr.,  1  wk.,  1  sq.  ft,  $  1,  and 
1  Ib.  have  been  said  to  contain  a  certain  number  of  units.     Thus,  1  da.  con- 
tains 24  units  of  1  hr.  ;  1  hr.,  60  units  of  1  min.  ;  1  wk. ,  168  units  of  1  hr. 
Any  measured  quantity  may,  however,  be  represented  by  any  convenient 
number  of  units.     Thus  1  da.  may  be  measured  by  75  units,  $  1  by  24  units, 
a  piece  of  work  by  40  units,  and  so  on. 

110.  (1)  Express  9  yd.  as  eighths  of  a  yd. 

A  I  I  I  I  I  I  I  7? 

Let  the  line  AB  be  drawn  to  represent  1  yd.    Think  of  1  yd.  as  contain- 
ing 8  units  of  length,  as  shown  in  the  diagram. 

Then  9  yd.  will  contain  9x8,  or  72  of  these  units. 
Therefore  9  yd.  is  equal  to  -7¥2-  of  1  yd. 

(2)  Express  $  -f-  as  a  fraction  with  20  as  a  denominator. 

Think  of  $  1  as  containing  20  units  of  5?  each.     Then  $  f  contains  f  of 
20,  or  15  of  these  units.     Therefore,  $|  = 


FRACTIONS  105 

Exercise  66 

Express  as  fractions : 

1.  $  8  as  lOths.  7.  5  yr.  as  12ths. 

2.  $4  as  20th s.  8.  3  da.  as  24ths. 

3.  9  yd.  as  ods.  9.  4  min.  as  GOths. 

4.  6  ft.  as  12ths.  10.  6  hr.  as  GOths. 

5.  14  gal.  as  4ths.  11.  8  pk.  as  Sths. 

6.  11  wk.  as  7ths.  12.  12  bu.  as  4ths. 

13.  In  each  of  these  questions  name  the  direct  measuring  unit 
in  your  fraction  as  an  actual  unit  of  measure  in  common  use. 

14.  Express  as  fractious  with  100  as  denominator:  1,  i,  i,  TL, 
i    JL   3    3.   2.  _o_ 

20'    25'    4'    5'    5'    1  0* 

Reduce,  illustrating  your  work  by  diagrams : 

15.  I  ft.  to  12ths.  19.  I  da.  to  24ths. 

16.  |  yd.  to  33ds.  20.  T%  to  TSths. 

17.  |  Ib.  to  IGths.  21.  T\  to  45ths. 

18.  f  gal.  to  28ths.  22.  }i  to  54ths. 

23.  What  are  the  new  units  of  measure  in  your  results  ? 
Where  possible,  identify  them  with  actual  units  in  common  use. 

111.    (1)  Express  8  in.  as  a  fraction  of  1  ft. 

8  in.  =  ^  or  f  of  a  ft.  =  f  ft. 

(2)  A  man's  capital  is  represented  by  20  units  of  value. 
He  invests  \  of  it  in  land  and  1  of  the  remainder  in  bank 
stock.  How  many  units  did  he  invest  in  bank  stock,  and 
what  part  is  it  of  his  entire  capital? 

The  amount  invested  in  land  =  \  of  20  units,  or  5  units. 

The  remainder  =  20  —  5,  or  15  units. 

The  amount  invested  in  stock  =  \  of  15,  or  5  units. 

.•.  the  amount  invested  in  stock  =25o»  or  i  °f  his  entire  capital. 


106  ARITHMETIC 

Exercise  67 

1.  Express  as  a  fraction  of  a  ft. : 

3  in.,  4  in.,  5  in.,  6  in.,  9  in.,  10  in. 

2.  Express  as  a  fraction  of  a  Ib.  Avoir. : 

14  oz.,  6  oz.,  9  oz.,  12  oz.,  15  oz. 

3.  Express  as  a  fraction  of  a  mo. : 

12  da.,  15  da.,  18  da.,  20  da.,  25  da. 

4.  If  1  mi.  is  the  measuring   unit,  what  number   measures 
each  of  the  following : 

20  rd.  ?   35  rd.  ?   80  rd.  ?   120  rd.  ?   150  rd.  ? 

5.  Express  as  a  fraction  of  a  da. : 

6  hr.,  9  hr.,  12  hr.,  15  hr.,  18  hr. 

6.  Express  as  fractions  of  a  dollar : 

25,  50,  75,  60,  80,  and  100  cents. 

7.  If  $  4  is  used  as  a  measure  of  $4,  what  is  the  number  ex- 
pressing the  measurement  ?     If  $  4  is  used  as  a  measure  of  $  3, 
what  is  the  number  ? 

8.  If  cloth  is  40^  a  yd.,  how  much  can  I  buy  for  30  ^  ? 

9.  If  oranges  cost  30^  a  doz.,  what  part  of  a  doz.  can  I  buy 
for  25  ^  ?     How  many  oranges  ? 

10.  A  man  walks  a  certain  distance  in  4  hr.     What  part  of 
it  does  he  walk  in  1  hr.  ?     In  2  hr.  ?     In  1  an  hr.  ? 

11.  A  can  do  a  piece  of  work  in  3  hr.,  B  the  same  amount  in 
4  hr.     If  the  work  be  measured  by  12    units,   how  many  units 
will  A  do  in  an  hr.  ?     How  many  B  ?     How  many  both  working 
together  ?     What  part  will  their  joint  work  for  an  hr.  be  of  the 
whole  work  ? 

12.  A  received  a  certain  sum  of  money,  B  twice  as  much,  and 
C  as  much  as  A  and  B  together.     How  many  units  measure  the 
entire  amount  ?     B  receives  what  part  of  the  whole  sum  ? 

13.  Given  that  pure  water  contains  15  parts  by  weight  of  oxy- 
gen, and  2  parts  of  hydrogen,  what  part  of  the  weight  of  a  gallon 
of  water  is  hydrogen  ? 


FRACTIONS  107 

14.  Six  brothers  join  in  paying  a  debt  of  $  700.     The  eldest 
pays  f  of  it,  and  each  of  the  others  ^  of  the  remainder.     How 
much  does  the  eldest  pay  ?     How  much  does  each  of  the  five  pay  ? 
This  is  what  part  of  the  whole  debt  ? 

15.  If  a  pipe  empties  a  tank  at  the  rate  of  12  gal.  in  1  min., 
what  is  the  rate  per  sec.  ? 

16.  $  40  is  divided  among  A,  B,  and  C,  giving  A  f  of  it,  B  f , 
and  C  the  remainder.     What  is  the  sum  of  A  and  B's  shares  ? 
What  is  C's  share  ?     C's  share  is  what  part  of  the  whole  sum  ? 

17.  The  value  of  a  mine  is  represented  by  10  units  of  money. 
A  man  who  owns  f  of  it  sells  f  of  his  share.     How  many  units 
did  he  own  ?     How  many  did  he  sell  ?     What  part  of  the  whole 
mine  did  he  sell  ? 

18.  The  length   of   an  oblong   is  8  ft.,  and  the  width  4  ft. 
What  part  of  the  perimeter  is  the  length  ? 

19.  An  oblong  6  ft.  wide  and  8  ft.  long  is  divided  into  strips, 
each  1  ft.  wide,  made  by  drawing  lines  parallel  to  the  length. 
What  part  of  the  area  of  the  oblong  is  the  area  of  one  strip  ? 

20.  The  area  of  an  oblong  is  16  sq.  ft.     What  part  of  its  area 
is  that  of  a  square  whose  side  is  2  ft.  ? 

21.  If  20  units  measure  the  cost  of  a  Ib.   of  tea  sold  at  a 
gain  of  -2%  of  the  cost,  how  many  units  are  gained  by  selling  ? 
What  is  the  selling  price  ?     The  cost  price  is  what  part  of  the 
selling  price  ?     What  is  the  ratio  of  the  selling  price  to  the  cost  ? 
The  gain  is  what  part  of  the  selling  price  ? 

22.  A  has  12  marbles,  and  B  has  3.     They  play  together,  and 
A  loses  J-  of  his  marbles.     How  many  has  B  now  ?     What  part 
are  they  of  what  A  now  has  ? 

23.  The  value  of  a  house  is  measured  by  5  units.     The  lot  on 
which  it  stands  is  worth  i  as  much  as  the  house.     What  is  the 

o 

measure  of  the  value  of  the  house  and  lot  ?     The  value  of  the  lot 
is  worth  what  part  of  both  together  ? 


108  ARITHMETIC 

24.  A  and   B   set  out  at  the   same  time  from   places  42  mi. 
apart,  and  meet  at  the  end  of  6  hr.     A  travels   at  the   rate  of 
3  mi.  an  hr.     How  far  does  B  travel  ?     What  is  B's  rate  ?     A's 
rate  is  what  part  of  B's  ? 

25.  Apples  are  sold  at  the  rate  of  12  for  a  dime,  and  bananas 
at  the  rate  of  8  for  a  dime.     Compare  the  value  of  an  apple  with 
that  of  a  banana. 

NOTE.  —  Let  the  dime  be  measured  by  24  units. 

26.  A  crew  can  row  6  mi.  an  hour  in  still  water.     What  is  the 
rate  of  rowing  up  stream  in  a  current  running  at  the  rate   of 
2  mi.  an  hr.  ?     What  is  the  rate  down  stream  ?     What  is  the 
ratio  of  the  rate  down  stream  to  that  up  stream  ? 

REDUCTION  OF  FRACTIONS 

112.    A  fraction  is  in  its  lowest  terms  when  its  numerator 
and  denominator  have  no  common  factor. 
(1)    Reduce  If^f  to  its  lowest  terms. 

The  object  of  reduction  is  to  give  a  more  definite  idea  of  the  value  of  the 
quantity  by  expressing  the  ratio  in  the  smallest  numbers. 


The  common  factors  are  3,  3,  and  5. 

The  effect  of  dividing  each  term  of  the  first  fraction  by  3  is  to  make  each 
measuring  unit  3  times  as  large,  and  to  reduce  the  number  of  these  units 
to  one-third  as  many.  Similarly,  with  the  second  division  by  3,  and  the 
third  by  5. 

(2)    Reduce  -J^f  to  its  lowest  terms. 


713 

558 

992 
713 

155 
124 

279 
155 

31 

124 

124 

71 3  _  713  -f-  31  _  23 
992  ~  992  -=-  31  "  32* 


FRACTIONS  109 

In  this  instance  the  direct  measuring  unit  of  the  fraction  f  f  is  31  times 
as  great  as  that  of  the  fraction  |i|,  but  only  JT  as  many  units  are  needed  to 
measure  the  quantity  denoted  by  |i|  or  f  f  of  the  primary  unit. 

(3)  A  real  estate  dealer  bought  a  house  for  $2545  and 
sold  it  for  $1679.  Find  the  ratio  of  the  selling  price  to  the 
cost  price. 

The  selling  price  =  if  £f  of  the  cost  price 

=  f|  of  the  cost  price. 
The  G.  C.  M.  of  1679  and  2545  is  73. 

Dividing  both  numerator  and  denominator  by  73,  we  have  the  required 
ratio  equal  to  f  f  . 

Exercise  68 

Reduce  to  its  lowest  terms  : 
1.    $£f.  6.    ff.  11.    ff 


o  <tt   1  6  714  1084 

»'  '  3T-  '*  6"3'  L*'  "96"' 

q  <J£  14  Q  32  1  Q  24 

3.  fy  2T'  °'  5~6'  ±<i5>  8~0' 

4  $  3.9.  Q  i_8  14  _48 

*•  ^48'  D'  99* 

5.  $.  10.  -. 


15.  If  I  pay  $  60  for  a  bicycle  and  afterward  sell  it  for  $  45, 
what  part  of  the  cost  do  I  sell  it  for  ? 

16.  A  merchant  sells  55  yd.  of  cloth  from  a  piece  containing 
66  yd.     What  part  of  the  whole  piece  does  he  sell  ? 

17.  .  A  man  buys  a  horse  for  $  128  and  sells  it  for  $  112.     Find 
the  selling  price  as  a  fraction  of  the  cost. 

18.  On  an  investment  of  $  88  a  merchant  gains  $  33.     Compare 
the  gain  with  the  cost. 

19.  Sixty  days  is  what  fraction  of  a  year  ? 

20.  Out  of  a  farm  containing  135  A.,  81  A.  were  sold.     Find 
what  part  of  the  farm  was  sold. 


HO  ARITHMETIC 

Exercise  69 

Eeduce  to  the  lowest  terms  : 

1         234  q         459  K       .3.9. 9J5.  «y        1218 

•*••       315'  1139*  °'       5661*  ''      TO¥T' 

2'      1T6TJ'  ^'      "5T9T"  6.       i  377-  8.      -go-g-g-. 

9.    A  person  made  $  282  on  an  investment  of  $  329.     What 
part  of  the  cost  did  he  gain  ? 

10.  A  speculator  paid  $  1071  for  a  lot  and  shortly  after  sold  it 
at  an  advance  of  $  259.     Find  his  gain  as  a  fraction  (in  its  lowest 
terms)  of  the  cost  price. 

11.  A  man  insured  his  house  which  cost  $  2552  for  $  1798. 
For  what  part  of  the  cost  did  he  insure  it  ? 

12.  I  paid  $  1449  for  a  house  and  insured  it  so  that  in  case  of 
fire  I  should  lose  only  $  184.     What  fraction  of  the  cost  would  I 
lose  in  case  of  fire  ? 

13.  Property  to  the  value  of  $  3341  was  assessed  for  $  1542. 
Find  the  ratio  of  the  assessed  valuation  to  the  real  value  of  the 
property. 

14.  Having  $  5943,  I  invested  $  5094  in  business.     What  part 
of  my  money  did  I  invest  in  business  ? 

113.  Such  an  expression  (as  $  21,  e.g.)  denotes  a  quantity 
in  which  two  units  of  measure  of  different  values  have  been 
used :  a  primary  unit  and  parts  of  this,  a  derived  unit. 

$21  is  the  number  7  in  disguise.  To  make  it  number  in 
the  strict  sense,  we  must  express  the  quantity  in  the  smaller 
unit  of  measure,  i.e.  as  $-J;  this  as  a  quantity  can  be  counted; 
$  2-i-  cannot  be  counted. 

o 

Aii  improper  fraction  as  an  expression  of  measured  quantity 
is  one  whose  numerator  is  equal  to  or  greater  than  its  de- 
nominator ;  as  -,  ,  ,  J°-. 


FRACTIONS  111 

(1)  Reduce  to  an  improper  fraction  75f  yd. 

Let  1  yd.       =     3  units  of  length  of  ±  yd.  each. 
Then  75  yd.  =  225  units  of  length  of  |  yd.  each. 

and  |  yd.  =      2  units  of  length  of  i  yd.  each. 

.'.  75|  yd.  =  227  units  of  length  of  1  yd.  each,  or  a|i  yd. 
Hence  if  we  count  the  smaller  unit  i  yd.  227  times,  we  measure  the  whole 
quantity  75  f  yd. 

Exercise  70 

Reduce  to  improper  fractions  : 

1.  $41                            8.    20}  gal.  15.  3^. 

2.  $4|.                            9.    Sfpk.  16.  20f. 

3.  $7f.                         10.    5T^lb.  17.  21}. 

4.  $161                         11.    12f.  18.  19f 

5.  $9fV                        12.    9f.  19.  7T%. 

6.  8J  yd.                      13.    12T2T.  20.  40J. 

7.  5&ft.  14.    6TV 

21.  What  are  the  two  units  of  measurement  in  each  question  ? 
What  is  the  unit  of  measurement  in  the  result  ?     How  many  of 
these  units  must  be  counted  to  measure  the  quantities  ? 

22.  What  is  the  quantity  which  is  measured  by  the  unit  |  ft. 
and  the  number  9  ?     If  the  same  quantity  be  measured  by  the 
number  3,  what  is  the  measuring  unit  ? 

23.  If  9  boys  receive  $|-  each,  what  sum  was  divided?     If 
the  same  sum  had  been  divided  among  4  boys,  what  would  each 
have  received  ? 

Exercise  71 

Reduce  to  improper  fractions : 

1.  15^-.                 5.    34T\.  9.  64ff  13.    29^. 

2.  21TV       6.  48if  10.  152|.  14.  STfff 

3.  29|f.       7.  51||.  11.  98T\.  15. 

4.  41^V       8.  36||.  12.  168|.  16. 


112  ARITHMETIC 

114.    (1)  Which  is  greater,  f  ft.  or  f  ft,  ? 

Consider  the  ft.  as  having  42  measuring  units,  then  |  of  this  is  35,  and 
f  is  36  units. 

.-.  f  ft.  is  greater  than  j  ft.  by  1  unit,  or  J2  ft. 

(2)  Compare  the  quantities  $J,  $^,  if. 

Let  $  1  be  represented  by  24  units  of  value.  Then  $  f ,  $  ^ ,  $  f ,  are  respec- 
tively equal  to  18,  14,  and  15  units.  Hence  the  first  fraction  is  the  greatest 
and  the  second  the  least. 

Exercise  72 

Find  the  greatest  and  least  of  the  following : 

1.    |  yd.,  f  yd.  4.    |  ft.,  f  ft.,  H  ft. 

2-    $$.  5.          8i 

3. 


7.    State  how  to  find  the  greatest  and  least  of  a  number  of 
fractions. 

115.    Express  -2^3-  yd.  in  terms  of  the  primary  and  derived 
units  of  measure. 

The  primary  unit  1  yd.  contains  5  derived  units  of  i  of  a  yd.  each. 
Hence  253-  yd.,  i.e.  23  derived  units  =  4  primary  and  3  derived  units  ; 

=  4f  yd. 
Or,  more  simply,  5)23 


Exercise  73 

Express  in  terms  of  the  primary  and  direct  measuring  units  : 

1         <K  43  c        187    TTT.  0385 

1.  3P  -5--.  -Y2~  yi.  y.    -g-g-- 

2.  $-8/.  6.    Wft-  10-    -fT 

3.  11^  da.  7.    -W-yd.  11.    -4TV 


4.    124         e  >      __        >  >    ____ 

13.    What  are  the  two  units  of  measurement  in  the  answers  to 
the  above  problems  ? 


FRACTIONS  113 

Exercise  74 

Reduce  to  integers  and  proper  fractions : 
1.    S.       2.    AA.       3.  .      4.    *l.'     5.    4.       6.    -, 


ADDITION  OF  FRACTIONS 

116.    The  sum  of  3  climes  and  4  dimes  =  7  dimes. 
The  sum  of  $  -f$  and  $  -^  =  $  j7^. 
The  sum  of  5  oz.  and  8  oz.  =  13  oz. 
The  sum  of  ^g-  Ib.  and  T8g-  Ib.  —  ||  Ib. 

Here  the  direct  measuring  unit  of  value  is  $  ^  or  1  dime, 
that  of  weight  ^  Ib.  or  1  oz. 

(1)  Add  1  ft.,  |  ft,,  |  ft. 

The  L.  C.M.  of  the  denominators  is  seen  to  be  12  ;  the  ft.  is  considered  as 
measured  by  12  equal  parts  or  units. 

\  ft.  =  6  units  ;  f  ft.  =  8  units  ;  f  ft.  =  9  units. 
The  sum  ==  6  +  8  +  9  or  23  units 

=  f|  ft.  or  \\\  ft. 
.•.  the  sum  =  1}^  ft. 

Prove  the  correctness  of  this  result  by  drawing  a  line  and  measuring  off 
i  ft.,  |  ft.,  and  |  ft.,  and  also  \\\  ft. 

(2)  Find  the  sum  of  $21f,  $15f,  $13$,  $8^. 

The  L.  C.  D.  =  72,  .  •.  the  dollar  is  considered  as  measured  by  72  units. 
$  !  -I-  if  £  +  $  4  +  $  _?_  _  27  +  60  +  32  +  42  or  161  units,  i.e.  $  -W  or  $  2fl- 
$  21  +  $  15  +  $  13  +  $  8  =  $  57. 

.  •.  the  sum  =  $  57  +  $  2f|  =  $  59}|. 

Exercise  75 

Find  the  sum  of: 

1-    f  fj  $  \-  3.    T7g-  oz.,  T2¥  oz.  5.    -f-  wk.,  £  wk. 

o         5     ff        4     fj-  A         13    flo       1  8    rlo  fj        37    -pii  1-1       41    min 

&•          T^TT     i-U.i     T~TT    -L  Ll.  TC.          7T~T    CLCt.  <     '.T   f      Lltt/.  D.          Tr7T    UULJ.il. «     TTTT    UUillt 

i.  ^  '      X  A  ^"i  'w"l  Ow  '     D  U 

I 


114  ARITHMETIC 

• 

Exercise  76 

In  the  following  exercise  what  is  a  convenient  number  of  direct 
measuring  units  by  which  to  represent  the  primary  units  $  1, 
1  ft.,  1  lb.,  1  gal.,  and  so  on  ? 

Find  the  sum  of  : 

1         <K  1      <tt  1          O        <K  1      <ffi  3     <K  5          q        2  f  4-       3  f  4-         A         1    f  4-       3    f  4-       5   f  4- 
1.      «fl)  2",   «fl)  4.        6.      «p  2",   «ff>  -g)  *p  g-.        O.      -g-  lb.,    4-  11.        <*.      -g   11.,  ^  1L.,  -g-  It. 

Prove  your  answers  to  3  and  4  correct  by  measuring  with 
a  ruler. 

5.    f  lb.,  f  lb.,  if  lb.        6.    1  gal.,  |  gal.        7.    f  bu.,  $  bu.,  11  bu. 
8-    fyr.,  |fyr-  9-    |  da.,  |  da.,  f  da. 


Prove  your  results  correct  by  reducing  each  fractional  quantity 
to  a  lower  denomination  (as  $  ^  to  ct.,  |  ft.  to  in.,  f  lb.  to  oz., 
and  so  on)  and  also  your  result.  Then  add  the  integers. 

\ 
Exercise  77 

Find  the  sum  of  the  following  fractional  parts  of  any  unit  : 

1342  4-5-      -8-      1-3- 

*'     "8?     5"?    "3"'  12>     15?     20' 

07169  fi319.      11      3.1 

*'     T2>     2"T>     2"8'  °'      14?     24?     21?     42' 

111712  fi       3      7      13      14      15 

"  D<      8?     9?     1 


8?  33?  44' 

7   Q2   Q4   9Q7   1822   A  A  6 
7.  0-j,  Oj^,  ^0^-g-  ,    5-1-    0- 

Q   93   Q5   K2   O  5    *> 
°-  ^^f>  °6">  Jyj  ^TT?  w 

9.  23^,  32|f,  43ft, 

10.  13T%,  47^,  34 

11.  State  how  to  find  the  sum  of  two  or  more  (1)  proper  frac- 
tions, (2)  mixed  numbers. 

12.  Explain  clearly  the  principles  involved  in  finding  the  sum 
of  two  fractions, 


FRACTIONS  115 

SUBTRACTION  OF  FRACTIONS 
117.    (1)  Find  the  difference  between  J  hr.  and  -f^  hr. 

Let     1  hr.  =  30  units  of  time. 

Then  -I  hr.  =  25  units  and  T\  hr.  =  16  units. 

O  1  O 

.-.  the  difference  =  25  -- 16  or  9  units,  i.e.  /„  or  ^  hr. 

Prove  this  result  by  expressing  f,  T85,  and  -^  hr.  in  inin.,  and  taking  the 
difference  of  the  first  two. 

(2)  Find  the  value  of : 


-$4  =  $2. 
The  L.  C.  D.  =  144. 

Let     $  1    =144  units. 

Then  $  f  f  =  116  units  ;  $  £|  =  51  units. 

116  -  51  =  65. 

7   —   Cj>O65 
" 


(3)  Subtract  G^  da.  from  9|  da. 

Let  1  da.  =  24  units  of  time. 

Then  9f  da.  -  6r72  da.  =  3  da.  +  9  units  -  14  units 

=  2  da.  +  33  units  -  14  units 
=  2  da.  +  19  units 


Exercise  78 

Find  the  value  of  : 


4-   i  gal-  --i  gal. 

2.  ^  ft.  -A  ft.  5.    flhr.-ifhr. 

3.  -J  pk.  -  -  1  pk.  6.   ^-j  da.  -  -  -^  da. 

7.  What  is  the  direct  unit  of  measure  in  each  question  ?     Ex- 
press your  answer  in  two  ways. 

8.  Prove  results  by  reducing  each  fraction  to  the  next  lower 
denomination  and  then  subtracting. 


116  ARITHMETIC 

Exercise  79 

Find  the  difference  as  a  fraction  of  any  unit  of  measure  : 

1         3  _  _  2  Q        6.-  -  .2  K        11       _   8 

J..  4  -g.  O.  ,7  g.  J2  g. 

2-      i  ~  6"'  4-      To"      •  TO"'  6.     T8"  ~~  12' 

7.  Give  the  three  steps  required  in   subtracting  one  proper 
fraction  from  another. 

8.  12  -f  13.    12-SJf.  18.    5ii  -3fJ. 

9.  5-TV  14.    8&-5TV  19. 

20 


11.    7-4¥V  16.    18}i  -l^V          21.    95|-84f. 


12.    9-6Jf  17. 


Exercise  80 

1.  -|  of  the  value  of  a  horse  =  $  60. 
i  of  the  value  of  a  horse  = 

-|  of  the  value  of  a  horse  = 
.-.  the  horse  is  worth  f 

Fill  out  the  blanks. 

2.  If  |  of  the  value  of  a  farm  is  $  9000,  what  is  the  value  of  \ 
of  the  farm  ?     What  is  the  farm  worth  ? 

3.  A  person  sold  a  cow,  gaining  ^  of  the  cost  price.     If  he 
gained  $  12,  what  did  the  cow  cost  him  ? 

4.  A  man  lost  in  business  |  of  his  property.     His  loss  was 
$  4500  ;  what  was  his  property  worth  ? 

5.  A  boy  lost  |  of  his  marbles,  and  then  had  60  marbles  left. 
What  fraction  of   his  marbles  did  he  have  left  ?     How  many 
marbles  had  he  at  first  ? 

6.  If  |  of  the  cost  of  a  gal.  of  a  wine  is  $  3,  what  was  the 
cost? 


FRACTIONS  117 

7.  A  merchant  sold  potatoes  at  75^  a  bu.,  gaining  J  of  the  cost. 
The  selling  price  was  what  fraction  of  the  cost  ?    What  was  the 
cost  price  per  bu.  of  the  potatoes  ? 

8.  A  merchant  sold  cloth  for  80^  a  yd.,  thereby  losing  1  of  the 
cost.     Find  the  cost. 

MULTIPLICATION  OF  FRACTIONS 

118.    (1)  Find  the  cost  of  12  yd.  of  cloth  at  If  per  yd. 

We  are  required  to  find  the  quantity  measured  by  the  number  12  and  the 
measuring  unit  $  |. 

12  yd.  cost-1/  x  $f  ==$9. 

(2)  Find  the  cost  of  f  yd.  at  $12  a  yd. 

The  cost  is  measured  by  the  number  -|  and  the  unit  $  12. 

f  yd.  costs  f  of  $12  -  $10. 
Explain  the  solution  j?  yd.  costs  12  x  f  £  =  $10. 

(3)  Find  the  area  of  a  floor  12  ft.  long  and  9|  ft.  wide. 

The  area  is  measured  by  the  unit  9f   sq.  ft.,   which  is  the  area  of  1 
strip,  and  the  number  12. 

The  area  of  1  strip  =  9|  or  3T9  sq.  ft. 

.-.  the  total  area  =  -\2-  x  \9  sq.  ft.  =  117  sq.  ft. 

(4)  Reduce  %  ft.  to  in. 

4 

§  ft.  =  §  x  £2  in.  =  —  in.  =  101  in. 
9  £       1  3 

3 

Exercise  81 

Find  the  cost  of : 

1.  10  yd.  of  cloth  at  $f  per  yd. 

2.  12  yd.  at  f  1 J  per  yd. 

3.  9yd.  at  $2J  a  yd. 

4.  f  yd.  at  $  6  a  yd. 

5.  If  the  cost  price  is  measured  by  the  number  -|  and  the  unit 
f  5,  find  the  cost. 


118  ARITHMETIC 

Reduce  to  in. : 
6.    f  ft.  7.    2fft.  8.   5fft. 

Reduce  to  yd. : 
9.    8  rd.  10.    12  rd.  11.    17  rd. 

Reduce  to  qt. : 

12.   of  gal.  13.   12ii  gal. 

Find  the  areas  of  the  following  rooms : 

14.  Length  24  ft.,  width  14  ft.  10  in. 

15.  Length  14  ft.,  width  12  ft.  6  in. 

16.  Length  18  ft.,  width  14  ft.  10  in. 

17.  Length  19  ft.,  width  16  ft.  4  in. 

18.  Length  22  ft.,  width  16  ft.  9  in. 

19.  Length  28  ft.,  width  20  ft.  11  in. 

Find  the  area  of  the  walls  of  a  room  whose : 

20.  Perimeter  is  80  ft.,  height  9  ft.  6  in. 

21.  Perimeter  is  63  ft.,  height  10  ft.  8  in. 

22.  Perimeter  is  67  ft.,  height  8  ft.  9  in. 

119.    (1)  Find  the  cost  of  12f  yd.  of  cloth  at  $  If  a  yd. 
12|  yd.  cost  12|  x  $  If  =  *£  *  *¥  =  fl  *v*  =  9 1?H- 

(2)  Find   the   area  of   the  four  walls  of   a  room  whose 
perimeter  is  62  ft.  8  in.,  and  height  8  ft.  9  in. 

The  perimeter  =  62  ft.  8  in.  =  62 f  ft.  =  ifA  ft. 
The  height  =  8  ft.  9  in.  =  8|  ft.  =  -^  ft. 

47 

-v  ocx       or 

.  •.  the  area  =  -^-  x  -  ^  sq.  ft.  =  548^  sq.  ft. 

(3)  Reduce  f  rd.  to  yd. 

4 

*  rd.  =  §  x  5i  yd.  =  ?  x  —  =  —  or  4f  yd. 
99  9  9 


FRACTIONS  119 

Exercise  82 

Keduce  to  yd. : 

1.  -J  rd.,  21  rd.,  5T6r  rd.,  3^  rd. 

Find  the  cost  of  the  following : 

2.  81  yd.  at  $  31  a  yd. ;  4f  yd.  at  $  2f  a  yd. ; 

20|  yd.  at  $  71  a  yd. ;  5J  yd.  at  $  2|  a  yd. 

3.  21  f  Ib.  of  sugar  at  5^  a  Ib.  j 
131  Ib.  of  sugar  at  5-^  a  Ib. 

4.  17f  yd.  of  cotton  at  llj^  a  yd. 

5.  151  doz.  of  eggs  at  14|^  a  doz. 

6.  8iT.  of  hay  at  $  llf  a  T. 

7.  Find  the  area  of  the  floor  of  a  room  whose  dimensions  are : 
(1)  12  ft.  6  in.,  10  ft.  8  in. ;  (2)  16  ft.  4  in.,  11  ft.  3  in. ;   (3)  18  ft. 
8  in.,  15  ft.  3  in. 

8.  Find  the  area  of  the  four  walls  of  a  room  whose  perimeter 
and  height  are  respectively :  (1)  52  ft.  6  in.,  9  ft.  4  in.  5  (2)  63  ft. 
4  in.,  10  ft.  6  in. 

9.  On  J  of  a  field  I  planted  potatoes  ;  on  J  of  the  remainder  I 
sowed  wheat.     What  part  of  the  field  did  I  sow  with  wheat  ? 

10.  I  withdrew  from  the  bank  f  of  my  deposit  and  then  f  of 
the  remainder.     What  part  of  the  original  deposit  did  I  take  out 
the  second  time  ? 

11.  A  man  who  owns  -f^  of  a  ship  sells  -J-  of  his  share.     What 
fraction  of   his  former  share  does  he  still  own  ?    What  fraction 
of  the  ship  ?     If  he  had  sold  ^  of  his  share,  what  part  of   the 
ship  would  he  have  still  owned  ? 

12.  A  grain  dealer  invested  -|  of  his  money  in  wheat,  and  f  of 
the  remainder  in  oats.     What  part  of  his  money  did  he  invest  in 
oats  ?     If  he  invested  $  3000  in  oats,  how  much  did  he  have  at 
first  ? 

13.  The  owner  of  a  farm  valued  at  $12,000  sells  f  of  it  to 
one  man,  and  -^  of  the  remainder  to  another.     What  part  of  the 


120  ARITHMETIC 

farm  does  lie  sell  to  the  second  man,  and  what  should  he  get  for 
it? 

14.  Four  brothers  enter  into  partnership  ;  the  eldest  puts  in  ^ 
of    the  capital  and  the  others  the  remainder  in  equal  shares. 
What   part   of    the    entire    capital    does   each   of    the    younger 
brothers  put  in  ?     If  they  each  put  in  $  2000,  what  is  the  entire 
capital  ? 

15.  If  I  own  -|  of  -f  of  a  business,  what  part  do  I  own?     If  I 
sell  J  of  my  share,  what  part  of  the  business  do  I  still  own  ? 

16.  A  man  left  his  farm  to  be  divided  among  his  three  sons  ; 
the  oldest  got  80  A.,  the  second  -J-  of  the  farm,  and  the  youngest 
-§-  as  much  as  the  other   two.     Prove  that  the  farm  contained 
210  A. 

17.  If  the  loss  is  measured   by  the   number  -|  and  the  unit 
$  17^  tons,  what  is  the  loss  ? 

If  the  gain  is  measured  by  the  number  -J  and  the  unit  $  llf  , 
what  is  the  gain  ? 

120.    A  owns  a  farm  containing  81|  A.,  B  owns  96^-  A., 
and  C  64^  A.     How  many  A.  do  they  own  altogether? 

Here  we  are  required  to  find  the  whole  quantity  measured  by  the  parts, 
81f  A.,96TVA,64iiA. 
Let  1  A.  =  120  units. 

Then  f  A.  =  45  units;  ^  A.  =  70  units  ;  11  A.  -  88  units. 
.-.  f  A..  +  ^  A.  -f  }l  A.  =:  45  +  70  +  88,  or  203  units  =  f  §f  A.  -  1T8^  A. 

81  A.  +  96  A.  +  64  A.  =  241  A. 
.-.  the  sum  =  241  A.  +  1T«^  A.  =  242T8^  A. 


121.  A  sum  of  money  is  divided  among  4  persons.  The 
first  receives  J,  the  second  ^,  the  third  -J,  and  the  fourth  the 
remainder.  It  is  found  that  the  first  received  $700  more 
than  the  fourth.  Find  the  sum  received  by  each. 

Consider  the  sum  of  money  as  made  up  of  60  units. 

The  first  receives  \  of  60  or  20  units;  the  second  15,  and  the  third  12 
units. 


FRACTIONS  121 

The  three  receive  20  +  15+12  or  47  units. 
The  fourth  receives  60  -  -  47  or  13  units. 
The  first  receives  20  --  13  or  7  units  more  than  the  fourth. 

7  units  =  $700. 
1  unit   =  $  100. 

.-.  the  first  received  20  units  or  $2000,  the  second  $1500,  the  third  $1200, 
and  the  fourth  $  1300. 

Exercise  83 

1.  What  is  the  combined  weight  of  three  men,  the  first  of 
whom  weighs  125 J-  lb.,  the  second  147f  lb.,  and  the  third  175J  lb.  ? 

2.  A  grocer  has  three  bbl.  of  oil ;  the  first  contains  ISf  gal., 
the  second  24f ,  and  the  third  16|.     Find  how  much  there  was  in 
the  three  bbl. 

3.  If  three  crocks  contain  respectively  8J,  12f,  and  14  j-|  lb. 
of  butter,  how  much  do  the  first  two  contain  more  than  the  third  ? 

4.  Four  farms  join  each  other;  the  first  contains  125|-  A.,  the 
second  78f  A.,  the  third  96J  A.,  and  the  fourth  HOf  A.     Find 
the  total  area. 

5.  A  person  paid  $  165J  for  a  horse,  and  $  231  more  than  that 
for  a  carriage,  and   shortly   after  sold  them  at  a  loss  of  $  46f . 
What  was  the  selling  price  ? 

6.  What  fraction  subtracted  from  the  sum  of  j  and  ^  will  have 
unity  for  remainder  ? 

7.  If  |  of  a  school  term  exceed  1  of  it  by  13J  da.,  how  many 
da.  are  there  in  the  whole  term  ? 

8.  I  am  the  owner  of  4-  of  a  ship  worth  $  30,000,  and  sell  i  of 
the  ship.     What  part  of  it  will  then  belong  to  me,  and  what 
will  it  be  worth  ? 

9.  Add  together  the  greatest  and  least  of  the  fractions  f,  j 


4>  ~8>    1  2> 

i-J,  and  subtract  this  sum  from  the  sum  of  the  other  two  frac- 
tions. 


122  ARITHMETIC 

10.  A  man  invested  -J-  his  fortune  in  land,  J  of  it  in  bank  stock, 
^  in  provincial  debentures,  and  lost  the  remainder,  $8000,  in 
speculation.     What  was  his  fortune  at  first  ? 

11.  Explain  why  fractions  having  different  denominators  must 
be  altered  in  form  before  their  sum  or  difference  can  be  expressed 
by  one  fraction. 

12.  A  gentleman  sold  -f^  of  an  estate  to  one  person,  and  T57  to 
a  second.     What  part  of  the  estate  did  he  still  retain  ?     If  this 
is  worth  $  8346,  what  is  the  value  of  the  estate  ? 

13.  If  a  man  fills  ^  of  a  cask  with  brandy,  \  with  wine,  and  -J- 
with  water,  and  it  lacks  21J  gal.  of  being  full,  how  many  gal. 
will  it  contain  ? 

14.  Show  that  the  fraction  :  -  is  greater  than  -J  and  less 
than  f                                         12  +  14 

15.  One  person  expends  $5  for  coal  at  $  7  per  T. ;  and  another 
$6  at  $9  per  T.     Which  of  them  obtains  the  greater  quantity 
of  coal  ? 

16.  An  estate  worth  $  10,000  is  left  to  A,  B,  and  C ;  f  to  A,  | 
to  B,  and  the  remainder  to  C.     Find  C's  portion  and  its  value. 

17.  If  during  the  day  I  pay  out  -J-,  then  1,  next  y1^-,  and  lastly 
-Jg-  of  the  money  I  had  in  the  morning,  what  fraction  of  it  have  I 
left?    If  the  sum  left  amounts  to  $  1.54,  what  sum  had  I  at  first? 

18.  (a)  How  much  must  be  added  to  the  denominator  of  £,  that 
the  resulting  fraction  may  be  equal  to  T7^  ? 

(6)  How  much  must  be  subtracted  from  the  denominator  of  -fa 
that  the  resulting  fraction  may  be  equal  to  ^  ? 

19.  In  a  certain  subscription  list,  |  of  the  number  of  subscrip- 
tions are  for  $  5  each,  i  are  for  $  4  each,  ^  are  for  $  2  each,  1  are 
for   $  1    each,   and   the   remaining    subscriptions,    amounting   to 
$  10.50,   are   for   50^    each.      Find    the    whole   number   of   sub- 
scribers, and  the  total  amount  of  their  subscriptions. 


FRACTIONS  123 

20.  A  man  lost  J-  of  his  property  in  speculation  ;  lie  afterwards 
purchased  a  partnership  in  business  for  $  16,000,  and  had  still 
$  6000  left.     What  was  he  worth  at  first? 

21.  The   sum   paid    for  494    gal.  of   oil,  including  a   duty  on 
each   gal.   which  amounts  to  i  of  the  cost  price  of  a  gal.,   is 
$  1719.12.     Find  the  duty  on  each  gal. 

22.  A  house  and  lot  cost  $  3600  ;  the  value  of  the  lot  is  J  that 
of  the  house.     Find  the  value  of  each. 

23.  What  must  be  the  length  of  a  plot  of  ground,  if  the  breadth 
is  15|  ft.,  that  its  area  may  contain  46  sq.  yd.  ? 

24.  A  boy  gives  ^  of  his  marbles  to  A,  1  to  B,  and  the  rest  to 
C.     C  loses  20,  and  has  then  70  less  than  A.     How  many  had 
each  at  first  ? 

25.  A  has  $3  more  than  i  of  the  whole  of  a  sum  of  money;  B 
has  $  4  more  than  \  of  the  whole  ;  and  C  has  $  5  less  than  i  of 
the  whole.     Find  the  sum.  divided. 

26.  After  spending  $  10  less  than  -|  of  my  money,  I  had  $  15 
more  than  T37  of  it  left.     How  much  had  I  at  first  ? 

122.    (1)  Find  the  product  of  if  x  6|  x  7fJ. 
Reducing  to  improper  fractions  and  cancelling,  the  product 

8       19 


=       xx       = 

3        9      1         9 


(2)  Find  the  volume  of  a  solid  whose  dimensions  are  2| 
in.,  2|  in.,  and  4J  in. 

The  volume  =  25.  x  2|  x  41  cu.  in. 

o  o  & 

—  --/-  x  f  x  I  cu-  in- 

—  1.5-6  or  31^  cu>  in. 

The  smallest  unit  of  volume  is  1  cu.  in. 

The  next  larger  unit  of  volume  is  4J  cu.  in. 

The  largest  unit  of  volume  is  2|  x  4|  cu.  in.  =  12  cu.  in. 

How  many  units  of  each  magnitude  measure  the  solid  ? 


124  ARITHMETIC 

Exercise  84 

Find  the  volume  of  the  solids  whose  dimensions  are : 

1.  2  in.,  4  in.,  6J  in.  4.    f\  ft.,  2f  ft.,  4f  ft. 

2.  1J-  in.,  21  in.,  2f  in.  5.    f  ft.,  f  ft.,  ^  ft. 

3.  2TV  in.,  2&  in.,  4|  in.  6.    JV  yd.,  2T<y  yd.,  Si  yd. 

7.  What  are  the  units   of  volume  in  the   first   and    second 
questions  ? 

8.  How  many  units  of  each  magnitude  measure  the  solids  in 
the  first  and  second  questions  ? 

Find  the  product  of  : 

9.  fxfxA-  12-          X  2x131 
10.   AxfxfJ.  13.   ^ 

Ul  0    v    1    V   2  8  14       1 

•     YT  •  *  8"  X  TS"  14> 

123.  (1)  A  owns  f  of  a  ship  and  B  the  remainder,  and  |  of 
the  difference  between  their  shares  is  f  1500.  What  is  the 
vessel  worth  ? 

Represent  the  value  of  the  ship  by  20  units  of  money. 
A  owns  f  of  the  ship  or  8  units,  B  |  of  the  ship  or  12  units. 
The  difference  between  their  shares  =  12  —  8  or  4  units,     f  of  the  differ- 
ence between  their  shares  =  3  units. 

3  units  =  $  1500. 
1  unit   =       500. 
20  units  =10,000. 
.-.  the  vessel  is  worth  $  10,000. 

(2)  A  man  lost  \  of  the  value  of  his  horse  by  selling  it  for 
For  what  should  he  have  sold  it  to  gain  |  of  its  value? 

Let  the  value  of  the  horse  =  5  units  of  money. 

Then  the  loss  on  selling  =  2  units  of  money. 
The  first  selling  price  =  3  units  of  money. 

The  second  selling  price  =  7  units  of  money. 
.-.  the  second  selling  price  =  |  of  $  GO  =  $  140. 


FRACTIONS  125 

(3)  A  person  who  has  f  of  a  mine  sells  J  of  his  share  for 
$ 6000.     What  is  the  value  of  the  whole  mine? 

Let  the  value  of  the  mine  be  measured  by  20  units  of  money. 

|  of  the  mine  =  8  units. 
The  amount  sold  =  |  of  8  units  =  6  units. 
6  units  =     $  6000. 
1  unit   =         1000. 
20  units  =     20,000. 
.-.  the  value  of  the  mine  =  $  20,000. 


Exercise  85 

1.  A  grocer  buys  tea  at  64^  a  lb.,  and  sells  it  so  as  to  gain 
/» i 

-5  of  the  cost  price.     Find  his  receipts  on  6043  Ib. 

O-i 

2.  A  man  bought  a  horse  for  $  120,  which  was  $  30  less  than 
^  of  one  and  a  half  times  what  he  sold  him  for.     How  much  did 
he  make  on  the  sale  of  the  horse  ? 

3.  If  I  own  f  of  f  of  |  of  a  ship  worth  $  20,000,  and  sell  1  of 
the  ship,  what  will  the  part  I  have  left  be  worth  ? 

4.  The  owner  of  a  ship  which  was  valued  at  $  10,000  sells  -| 
of  it  for  $3800,  and  then  1  of  the  remainder  for  $1800.     What 
did  he  gain  or  lose  by  the  transaction  ? 

5.  A  man  has  $  4000  in  the  bank.     He  drew  out  -/-$  of  it,  and 
then  1-  of  the  remainder,  and  afterwards  deposited  i  of  what  he 
had  drawn  out.     How  much  had  he  then  in  the  bank? 

6.  A  man  divided  a  farm  among  three  sons ;  to  the  first  he 
gave  80  A.,  to  the  second  -J  of  the  whole,  and  to  the  third  f  as 
much  as  to  both  the  others.     How  many  A.  did  the  farm  con- 
tain ? 

7.  Divide  $65.80  between  two  persons,  so  that  one  shall  re- 
ceive one-third  as  much  again  as  the  other. 

8.  Five  brothers  join  in  paying  a  sum  of  money ;   the  eldest 
pays  |-  of  it,  and  the  others  pay  the  remainder  in  equal  shares. 


126  ARITHMETIC 

It  is  found  that  the  eldest  brother  pays  $300^^  more  than  a 
younger  brother's  share.     Find  the  sum  of  money. 

9.  Show  clearly  that  both  terms  of  a  fraction  can  be  multi- 
plied by  the  same  number,  without  changing  the  value  of  the 
fraction. 

10.  A  man  commenced  business  with  a  capital  of  $8000;  the 
first  year  he  gained  $40  for  every  $100   invested,  adding  his 
gain  to  his  capital ;    the  second  year  he  gained  $  25  for  every 
$  100  invested,  adding-  his  gain  as  before ;  the  third  year  he  lost 
i  of  his  accumulated  capital.     How  much  did  he  make  in  the 
three  years  ? 

11.  If  J  of  ^  of  an  A.  produces  43  bu.  of  potatoes,  how  many 
bu.  will  an  A.  produce  ? 

12.  A  paid  $  60  per  A.  for  his  farm,  which  was  J  as  much  as 
B  paid  per  A.  for  his  farm  of  150  A.     Find  the  entire  cost  of 
B's  farm. 

13.  If  the  sum  paid  for  247  gal.  of  spirits  amounts,  together 
with  the  duty,  to  $  859.56,  and  the  duty  on  1  gal.  be  1  part  of  its 
original  cost,  what  is  the  duty  per  gal.  ? 

14.  A    piece    of   cloth,    when    measured    with   a  yd.   measure 
which  is   |  of   an  in.    too    short,    appears  to   be   10J  yd.    long. 
What  is  its  true  length  ? 

15.  I  had  a  sum  of  money,  of  which  I  paid  away  J,  then  \  of 
the  remainder,  then  f  of  what  was  still  left,  and  found  that  I  had 
still  left  half  a  dollar  less  than  1  of  J  of  the  whole.     What  sum 
had  I  at  first  ? 

• 

16.  A  man  who  owns  £f  of  a  mill  sells  f  of  his  share.     What 
fraction  of  the  mill  does  he  still  own  ?    Had  he  sold  f  of  the  mill, 
what  fraction  of  the  mill  would  he  still  have  owned  ? 

17.  Find  the  sum  of  the  greatest  and  least  of  the  fractions  f, 
1%,  f,  and  2^0 ;  the  sum  of  the  other  two;  and  the  difference  of 
these  sums. 


FRACTIONS  127 

18.  If  when  -3-%  of  a  certain  time  has  elapsed,  and  then  1  hr.; 
and  then  |-  of  the  remainder  of  the  time,  it  is  found  that  16  min. 
of  the  time  still  remains,  what  was  the  whole  time  ? 

19.  A  man  sold  -J  of  his  wheat  and  then  -fa  of  the  remainder, 
and  next  \^  of  what  then  remained,  and  had  18  bu.  more  than  -f^ 
of  his  wheat  left.     How  many  bu.  had  he  at  first  ? 

20.  A  man  sold  ^  of  his  farm,  then  1  of  the  remainder,  then  ^ 
of  what  remained,  then  ^  of  what  still  remained,  and   he  then 
found  that   he   had   sold   altogether  72   A.   more   than   he   had 
remaining.     How  many  A.  had  he  at  first  ? 

DIVISION  OF  FRACTIONS 

124.  Divide  3J  ft.  by  f  ft. 

Here  we  are  given  the  whole  measured  quantity  3J  ft.  and  the  unit  of 
measure  f  ft.,  and  we  are  required  to  find  the  number  which  expresses  the 
ratio  of  the  whole  quantity  to  the  unit.  Consider  the  ft.  as  measured  by  6 
units  of  length.  Then  3i  ft.  contains  20  units,  and  f  ft.  contains  5  units. 

.-.  the  number  or  ratio  =  4. 

Prove  that  |  ft.  divides  3^  ft.  4  times  by  drawing  a  line  3i  ft.  or  40  in. 
long,  and  measuring  it  with  a  line  f  ft.  or  10  in.  long.  Prove  that  the 
quotient  4  is  correct  by  multiplying  |  ft.  by  4  and  finding  the  product  to 
be  3i  ft. 

125.  If  we  represent  any  quantity  by  -|  of  itself,  we  think 
of  the  quantity  as  made  up  of  5  units,  and  the  unit  as  equal 
to  jr  of  the  quantity.     Hence  we  may  think  of  any  quan- 
tity as  equal  to  5  x^  (i.e.  five  times  one-fifth)  of  itself. 

What,  then,  is  the  meaning  of  the  terms  5  and  ^  considered 
separately  ?  5  shows  the  relation  or  the  ratio  of  the  quantity 
to  the  unit  of  measure,  and  J  shows  the  ratio  of  the  unit  of 
measure  to  the  quantity.  Numbers  thus  mutually  related 
are  said  to  be  reciprocal. 

B 

A 1 1 1 1 1 Q 


128  ARITHMETIC 

Again,  let  the  line  AB,  which  is  divided  into  5  equal  parts,  represent 
the  primary  unit,  $1.  Then  AC  represents  the  quantity  denoted  by  $|. 
Hence  it  is  evident  from  the  diagram  that  f  is  the  ratio  of  AC  to  AB,  i.e.  of 
the  quantity  denoted  by  $  §  to  the  primary  unit,  $  1  ;  also,  that  its  recip- 
rocal |  is  the  ratio  of  AB  to  AC,  i.e.  of  the  primary  unit,  $1,  to  the 
quantity  denoted  by  $  f .  Similarly,  the  ratio  of  $  1  (which  contains  2 
units)  to  the  quantity  $f  (which  contains  3  units)  is  equal  to  f,  i.e.  to  the 
reciprocal  of  f . 

Exercise  86 

1.  How  many  units  of  money  are  there  in  $  1  and  also  in  $  }, 
the  unit  being  one-half  dollar  ? 

How  many  in  $  1  and  also  in  $  f ,  the  unit  being  one-third  of 
a  dollar  ? 

How  many  in  $  1  and  also  in  $  f ,  the  unit  being  one-fifth  of 
a  dollar  ? 

2.  What  is  the  ratio  of  $  1  to  f  f  ?    $|?    If?    $21?    $1? 

<H?    1    O       fit   4.   O 

<ff>  t  •      «B>  y  • 

3.  What  part  is  $  1  of  each  of  these  quantities  :  $  2  ?  $  -1/  ? 

$2_o?   $3^?   $4T2T? 

4.  What  is  the  ratio  of  1  Ib.  to  f  Ib.  ?   f  Ib.  ?   2f  Ib.  ?   2|  Ib.  ? 

5.  What  is  the  ratio  of  2  Ib.  to  each  quantity  given  in  ques- 
tion 4  ?     Of31b.  ?    Of41b.  ? 

6.  What  is  the  ratio  of  1J  Ib.  to  each  quantity  given  in  ques- 
tion 4? 

7.  If  |  of  a  yd.  cost  $  1,  what  part  of  $  1  will  1  yd.  cost  ? 

8.  If  |  of  a  yd.  cost  $  1,  what  part  of  $  I  will  1  yd.  cost  ? 

9.  If  f  of  a  yd.  cost  f  21,  what  part  of  $21  will  1  yd.  cost? 
How  much  will  1  yd.  cost  ? 

10.  If  31  yd.  of  cloth  cost  $lf,  what  will  1  yd.  cost  ? 

11.  If  41  Ib.  of  sugar  cost  26^,  what  will  1  Ib.  cost  ? 


FRACTIONS  129 

12.  If  f  of  a  yd.  of  cloth  costs  32^,  what  will  1  yd.  cost? 
What  will  3  yd.  cost  ?   41  yd.  ?   T9g  yd.  ? 

13.  If  71  Ib.  of  raisins  cost  85^,  what  will  41  Ib.  cost  ? 

126.  (1)  At  $5  a  yd.,  how  much  lace  can  be  bought  for 

f7    9 
TO  ' 

Here  we  are  given  the  whole  quantity  or  $  ^  ,  and  the  measuring  unit  $  5, 
and  we  are  required  to  find  the  number. 

Consider  $  1  as  measured  by  10  units. 

Then  $^  contains  7  units  and  $5  contains  50  units.  Therefore  the 
number  of  yd.  =  $  ^  -f-  $  5  =  7  units  -r-  50  units  =  -5^. 

(2)  Find  the  value  of  $f  -=-  If. 

Consider  f  1  as  made  up  of  28  units.  Then  $  f  is  equal  to  24  of  these 
units  and  $  f  to  35  of  these  units.  Therefore,  $  f  -4-  $  |  =  24  units  -^  35  units 
:=  ff.  Hence,  to  divide  one  fractional  quantity  by  another,  we  first  reduce 
them  to  the  same  unit. 

127.  It  may  be  observed  that  ||  is  equal  to  f  multiplied 
by  ^,  the  reciprocal  of  -|.      Hence,  to  divide  one  fraction  by 
another,  multiply  the  first  fraction  by  the  reciprocal  of  the 
second. 

Exercise  87 

In  the  following  exercise,  solve  (1)  by  reducing  the  quantities 
to  the  same  unit,  (2)  by  multiplying  by  the  reciprocal  : 

1       <fi  2     .    <tt  3  Q       5   -nlT-      .     1   -nV  K       7  _i_    5 

W  "5"  ~z~  <JP  x*  *'•      "9"  ]/*»••  ~-~  ~v    P"-«  O«      "ir  ^^  T~2~* 

2.    fyd.-iyd.  4.   21  -  f.  6.   ^  - 


128.    (1)  Reduce  2J  ft.  to  a  fraction  of  1  yd. 

1  ft.  -  i  yd. 
2i  ft.  =  2J  x  i  yd.  =  |  x  1  yd.  =  f  yd. 

Give  the  solution  which  follows  from  the  fact  that  2|  ft.  is  the  whole 
measured  quantity,  3  ft.  the  unit  of  measurement,  and  that  we  are  required 
to  find  the  number. 

K 


130  ARITHMETIC 

(2)  Find  the  number  of  strips  required  to  carpet  a  room 
12  ft.  wide  with  carpet  2£  ft.  wide. 

The  number  of  strips  =  12  ft.  -4-  2£  ft.  =  \2-  x  f  =  4f ,  i.e.  5. 
.  •.  5  strips  are  required. 

Exercise  88 


1.  Reduce  to  the  fraction  of  a  ft.  :  1^-  in.,  2J-  in.,  4J  in.,  2f  in. 

2.  Reduce  to  the  fraction  of  a  yd.  :  11  ft.,  2|  ft.,  11  ft.,  2|  ft. 

1,  f  371,  $  621,  $  87, 


3.    Reduce  to  fractions  of  $  100  :  $  121,  f  371,  $  621,  $  871 


4.  If  I  paid  $  ^  for  6  yd.  of  cloth,  what  will  1  yd.  cost  ? 

5.  If  I  paid  $4^-  for  14  yd.  of  cloth,  what  will  1  yd.  cost? 

6.  If  $  f  is  considered  as  measured  by  5  units,  what  is  the 
value  of  1  unit  ? 

7.  If  $  8f  is  measured  by  the  unit  $  5,  what  is  the  number 
expressing  the  measurement  ? 

8.  At  $  2J  a  yd.,  how  much  lace  can  be  bought  for  $  -f%  ? 

9.  At  $  3-J  a  yd.,  how  much  lace  can  be  bought  for  $  -^  ? 

10.  Find  the  number  of  strips  of  carpet  required  to  carpet  a 
room  23  ft.  4  in.  long,  with  carpet  2  ft.  wide.     With  carpet  3  ft. 
wide. 

11.  Find  the  lengths  of  the  following  rooms  : 

Area  466|  sq.  ft.,  width  20  ft. 
Area  2421  sq.  ft.,  width  15  ft. 
Area  681^  sq.  ft.,  width  25  ft. 
Area  8771  sq.  ft.,  width  27  ft. 

129.    (1)  Divide  If  ft.  by  f  ft. 

If  ft.  +  f  ft.  =  |  --  f  =  |  x  |  =  4|. 

Prove  this  by  drawing  a  line  1|  ft.  long,  and  dividing  it  into  parts  |  ft. 
long. 


FRACTIONS  131 

(2)  If  I  paid  $  J  for  f  of  a  yd.  of  cloth,  what  is  the  price 
per  yd.?  Here  we  are  given  the  whole  measured  quantity, 
that  is,  $-§-,  and  the  number,  or  -|,  and  we  are  required  to  find 
the  measuring  unit. 

f  of  a  yd.  costs  $  f. 

.-.  lyd.  costs  ff-4-|  =  $f  xf  = 


Exercise  89 

1.    Reduce  to  the  fraction  of  a  rd.  :  2f  yd.,  If  yd.,  4±  yd.,  6|  yd., 


2.  Find   and    prove   by    actual    measurement   the   following  : 
o  i  f  t  _•_  i  f  t  .  1  1  f  f  _._  i  f  f  .  9  1  f  f    •  i  f  t  •  1  7  f  t  _i_  i  f  t  •  4.  i  f  t   •    5  f  t  • 

--g-II.—  g-  II.  ,     L^It.  —  g-ID.  ,    ^TIl.  —  g-rt.  ,    J_-g  rC.  —  ^11.  ,    4  g-  It.  —  YZ  II"  J 

Glft.H-yLft. 
o  1  L 

3.  At  $3TL  a  bbl.,  how  much  flour  can  I  buy  for  $17i?     At 
$  9|-  a  yd.,  how  much  lace  can  be  bought  for  $  2^  ? 

4.  Find  the  number  of  strips  of  carpet  required  to  carpet  each 
of  the  following  rooms  : 

Width,  22  ft.  6  in.  ;  width  of  carpet,  2  J  ft.  (221  ft.  -j-  2|  ft.) 
Width,  24  ft.  3  in.  ;  width  of  carpet,  21  ft. 
Width,  8  yds.  2  ft.  ;  width  of  carpet,  -|  yd. 
Width,  9  yds.  1  ft.  ;  width  of  carpet,  -|  yd. 

5.  Find  the  length  of  each  of  the  following  rooms  : 

Area  of  floor,  126f    sq.  yd.  ;  width,  8  yd.  2  ft. 
Area  of  floor,  133£|  sq.  yd.  ;  width,  9f  yd. 
Area  of  floor,  334|i  sq.  ft.  ;  width,  15  ft.  9  in. 
Area  of  floor,  2531    sq.  ft.  ;  width,  10  ft.  8  in. 

6.  Find  the  perimeter  of  each  of  the  following  rooms  : 

AREA  OF  THE  WALLS  HEIGHT 

5331  sq.  ft.  8  ft.  4  in. 

704|  sq.  ft.  9  ft.  4  in. 

8771  sq.  ft.  10  ft.  9  in. 

65J  sq.  yd.  2  yd.  2  ft. 


132  ARITHMETIC 

7.  If  %  of  the  sum  a  man  paid  for  a  horse  is  $83^,  what  did 
the  horse  cost  ? 

8.  If  -f  of  a  yd.  of  cloth  cost  $  f  ,  what  will  1  yd.  cost  ? 

130.    What  is  the  ratio  of  8f  da.  to  IfJ  da.  ? 

The  ratio  of  8f  da.  to  1^  da.  =  8f  -4-  li*  =  ^  -5-  f  £  =  -6/  x  if  =  4f. 

Exercise  90 

1.  What  is  the  ratio  of  $  5|  to  $  3|  ? 

2.  What  is  the  ratio  of  144-  T.  to  6|  T.  ? 

O  O  O 

3.  If  6|  T.  of  hay  cost  $42,  what  part  of  $42  will  lif  T. 
cost  ?     How  much  will  l^f  T.  cost  ? 

4.  If  $  -^g-  is  represented  by  unity,  what  number  will  represent 

$429 
ft  • 

5.  Find  the  value  of  (1)  5|  -5-  3f  ;  (2)  7£  -5-  4yL  ;  (3)  2T5¥  -  5Jf  I 


131.    Simplify 


_  8 
- 


Or  thus,  multiplying  the  numerator  and  denominator  by  12,  the  L.  C.  M. 
of  3  and  4,  we  have 

41^56^8 
5    ~  63  ~  9* 


Exercise  91 

Simplify,  using  both  methods  : 

3     2f  5 

- 


6J-  8  '    17F 


2.     JL,  4.    rl.  6.        LA.  8.    J-L 

13  Q8  116 


FRACTIONS  133 

132.  (1)  A  man  earns  $4J  a  da.,  and  his  daily  expenses 
are  $1J.  How  many  da.  will  it  take  him  to  save  enough 
money  to  buy  a  bicycle  costing  858-J-? 

The  sum  saved  each  day  =  $41-  --  $  1|  :=  $2J$. 
.  •.  the  number  of  days  =  58  i  -=-  2£g-  ==  25. 

(2)  A  farm  of  340  A.  was  divided  between  two  sons,  so 
that  f  of  the  youngest  son's  share  was  equal  to  f  of  the  eldest 
son's  share.  Find  the  size  of  each  farm. 

|  of  the  youngest  son's  share  =  f  of  the  eldest  son's  share. 

The  youngest  son's  share  =  f  -*-  s  ,  or  -|  of  the  eldest  son's  share. 
...  9  _|_  |?  or  -1/-  of  the  oldest  son's  share  =  340  A. 

.  •.  the  oldest  son's  share  =  340  A.  -=-  11  =  180  A. 

.  •.  the  youngest  son's  share  =  f  of  180  A.  =  160  A. 

Complete  the  following  solution  of  question  (2)  : 

Let  |  of  the  youngest  son's  share  =  6  units. 


(3)  Sold  tea  at  90^  per  lb.,  having  gained  ^  of  the  cost. 
Find  the  selling  price  per  lb.  if  he  had  lost  f$. 

Let  the  cost  of  1  lb.  be  measured  by  20  units. 
Then  -^  of  the  cost  price  =    3  units, 
The  first  selling  price         =  23  units. 
The  second  selling  price    =  17  units. 
.  •.  the  second  selling  price   =  ij  of  the  first 

=  if  of  90^  =  06  J#. 

Exercise  92 

1.  A  vessel  holds  2^  qt.     How  many  times  can  it  be  filled 
from  a  barrel  containing  31^-  gal.  of  oil  ?     After  filling  the  vessel 
as  often  as  possible,  how  much  oil   will  remain  in  the  barrel  ? 
What  fraction  of  a  vesselful  will  this  remaining  quantity  be  ? 

2.  Divide  the  sum  of  the  fractions  f  and  T4g-  by  the  product 
of  ^  and  1J,  and  reduce  the  result  to  its  lowest  terms. 

3.  The  bottom  of  a  cistern  measures  7  ft.  6  in.  by  3  ft.  2  in. 
How  deep  must  it  be  to  contain  76  cu.  ft.  of  water  ? 


134  ARITHMETIC 

4.  A  man  sold  24  horses  for  $  150  each ;  on  half  of  them  he 
gained  -^  of  what  they  cost ;  and  on  the  remainder  he  lost  ^  of 
what  they  cost.     Find  his  whole  gain  or  loss. 

5.  By  selling  cigars  at  the  rate  of  $2.60  for  4  doz.,  it  was 
found  that  -f-  of  the  cost  was  gained.     Find  the  price  at  which 
each  cigar  ought  to  have  been  sold  in  order  to  gain  ^  of  the 
original  cost. 

6.  James  received  a  present  of  some  money.     He  gave  -J-  of 
it  to  his  sister,  and  -|  of  the  remainder  to  his  brother,  and  kept 
the  rest,  $  4,  for  himself.     How  much  did  he  receive,  and  how 
much  did  his  brother  get  ? 

7.  A  tree  of  140  ft.  in  length  was  broken  into  two  pieces  by 
falling,  and  -f^  of  the  longer  piece  was  equal  to  ^  of  the  shorter. 
Find  the  length  of  each  piece. 

8.  A  owns  -f-  of  a  ship  and  B  the  remainder,  and  f  of  the 
difference  between  their  shares  is  $1500.     What  is  the  vessel 
worth  ? 

9.  A  person  who  has  f  of  a  mine  sells  J  of  his  share  for 
$  6000.     What  is  the  value  of  the  whole  mine  ? 

10.  Divide  $  2380  between  A  and  B  so  that  -|  of  A's  share  will 

o 

be  equal  to  J  of  B's. 

11.  If  13  were  added  to  a  certain  number,  f  of  -J  of  the  sum 
would  be  40.     Find  the  number. 

12.  Three  partners,  A,  B,  and  C,  gain  $  17,100  ;  A's  gain  and 
C's  are  together  $  11,000,  and  f  of  A's  is  equal  to  -fo  of  C's.    Find 
each  man's  gain. 

13.  A   grocer   in  selling  goods  sells  15|  oz.  for  1  Ib.      How 
much  does  he  cheat  a  customer  who  buys  to  the  amount  of  f  40  ? 

14.  The  numerator  of  a  certain  fraction  is  -J-  as  much  again 
as  its  denominator,  and  the  sum  of  the  numerator  and  denomi- 
nator is  352.     Find  the  fraction. 

15.  A  cannon  ball  travels  at  the  rate  of  1500  ft.  in  1-J-  sec. 
How  far  will  it  have  gone  in  T^  of  a  min.  ? 


FRACTIONS  135 

16.  A  man  divides  the  value  of  his  estate  equally  among  his 
three  sons.     The  first  son  gains  an  amount  equal  to  1  of  what  he 
has ;  the  second  loses  -^  of  what  he  has,  and  the  third  gains  -|  of 
what  he  has,  and  then  loses  -J-  of  what  he  has  after  his  gain ;  and 
now  the  sons  together  have  $  300  less  than  the  value  of  the  estate. 
What  was  its  value  ? 

17.  Given,  that  pure  water  is  composed  of  oxygen  and  hydro- 
gen in  the  proportion  by  weight  of  15  to  2,   find  the  weight 
of   each   in   a   cu.   ft.    of   water.      (A   cu.    ft.    of   water   weighs 
1000  oz.) 

18.  A  line  A  is  half  as  long  again  as  B,  and  B  is  one-quarter 
as  long  again  as  C.     What  fraction  of  the  length  of  A  is  equal  to 
i  of  the  length  of  C  ? 

19.  A  and  B  go  into  business  with  equal  sums  of  money.     A 
gains  a  sum  equal  to  f  of  what  he  had  at  first,  and  B  loses  $  60. 
A  then  has  1^  times  as  much  as  B.    What  sums  had  they  at  first  ? 

20.  A  and  B  sit  down  to  play.    A  has  5^-  times  as  much  money 
as  B.     At  the  first  game  B  wins  4  of  A's  money.     What  fraction 
of  B's  money  must  A  win  back  so  that  they  may  have  equal 
shares  ? 

21.  A  young  man's  salary  increased  ^  every  year;  his  expenses 
each  year  were  1  of  his  salary,  and  at  the  end  of  4  years  he  had 
saved  $  1050.     Find  his  last  year's  salary. 

22.  Find  a  fraction  equivalent  to  ^V?  an(^  having  its  numerator 
44  less  than  its  denominator. 

23.  A  trader  spent  the  first  year  $  40,  and  added  to  his  capital 
\  of  what  he  had  left ;  the  second  year  he  spent  $  50,  and  added 
to  his  capital  1  of  what  he  had  left ;  the  third  year  he  spent  $  60, 
and  added  to  his  capital  1  of  what  he  had  left ;  he  finds  that  his 
capital  is  now  lii  of  what  it  was  at  first.     Find  the  original 
capital. 

24.  In  a  field  in  which  cows  and  sheep  were  grazing,  ^  of  the 
total  number  were  cows ;  but  when  3  cows  more  were  driven  in, 


136  ARITHMETIC 

the  latter  numbered  T2T  of   the  whole.     How  many  sheep  were 
there  ? 

25.  Two  persons,  A  and  B,  finish  a  work  in  20  da.,  which  B 
by  himself  could  do  in  50  da.     In  what  time  could  A  finish  it  by 
himself?      How  much   more   of   the  work  is  done   by  A  than 
byB? 

Let  the  work  be  represented  by  100  units  of  work. 

A  and  B  do  100  units  -^  20  or  5  units  in  1  da. 
B  does  100  units  H-  50  or  2  units  in  1  da. 
A  does  5  —  2  or  3  units  in  1  da. 
.-.  A  does  the  work  in  100  -=-  3  or  331  da. 
Again, 

A  does  3  —  2  or  1  unit  more  than  B  in  1  day. 

A  does  20  units  more  than  B  in  20  days. 
/.  A  does  T^TT  or  i  of  the  work  more  than  B. 

1  U  U  O 

26.  If  A  can  do  a  piece  of  work  in  3  da.,  and  B  in  4  da.,  in 
what  time  Can  both  working  together  do  the  work  ? 

27.  A  alone  can  do  a  piece  of  work  in  11  da.,  and  B  alone 
can  do  it  in  17  da.     How  long  would  they  take  to  do  it  to- 
gether ? 

28.  A  and  B  can  do  a  piece  of  work  together  in  6  da,,  and  B 
can  do  J  of  the  same  in  11  da.     How  long  would  each  be  in 
doing  it  alone  ? 

29.  A  can  do  a  piece  of  work  in  5,  B  in  6,  and  C  in  8  da. 
If  A  and  B  work  at  it  2  da.  each,  how  long  will  it  take  B  and 
C  to  finish  it  ? 

30.  A  and  B  can  do  a  piece  of  work  in  3  da.,  B  and  C  in  6 
da.,  and  A  and  C  in  4  da.     If  $  16  be  paid  for  the  work,  what  is 
each  man  worth  per  da.  ? 

31.  A  and  B  can  do  a  piece  of  work  alone  in  15  and  18  da. 
respectively;  they  work  together  at  it  for  3  da.,  when  B  leaves, 
but  A  continues,  and  after  3  da.  is  joined  by  C,  and  they  finish 
it  together  in  4  da,     In  what  time  would  C  do  the  work  by 
himself? 


FRACTIONS  137 

G.  C.  M.  AND  L.  C.  M.  OF  FRACTIONS 

133.  To  find  the  G.C.  M.  of  5J  and  2f-  of  any  unit. 

Consider  the  unit  as  divided  into  15  units,  then  5|  is  equal  to  80,  and  2| 
to  36  of  these  units.  Here  the  G.  C.  M.  is  equal  to  the  G.  C.  M.  of  80  and  36, 
or  to  4  of  these  units,  i.e.  to  T4-  of  the  primary  unit. 

Hence  we  have  the  following  rule  : 

To  find  the  Gr.  C.  M.  of  a  number  of  fractions,  reduce  the 
fractions  to  a  common  unit,  and  then  find  the  Gr.  C.  M.  of  the 
resulting  numbers.  Express  the  result  as  a  fraction  of  the  pri- 
mary unit. 

134.  To  find  the  L.  C.  M.  of  5J  and  2f  . 

Reduce  the  fractions,  as  in  the  preceding  paragraph,  to  a  common-  unit. 
The  L.  C.  M.  of  80  and  30  is  720,  and  the  L.  C.  M.  is  equal  to  "T25°,  or  48  of 
the  primary  unit. 

Hence,  to  find  the  L.  0.  M.  of  a  number  of  fractions,  reduce 
them  to  a  common  unit,  and  then  find  the  L.  C.  M.  of  the  re- 
sulting numbers.  Express  this  result  as  a  fraction  of  the  pri- 
mary unit. 

Exercise  93 
Find  the  G.  C.  M.  of  : 


1.    ifand^L0-.  2.    |  and  If.  3.    21£  and  7 

Find  the  L.  C.  M.  of  : 
4.    f,  |,  and  -jL  5.    11  21  and  3f.         6.    4|,  if,  and  4f. 

7.  A  man  has  a  triangular  field,  of  which  the  sides  are  115  J 
ft.,  12S^r  ft.,  and  134J  ft.     Find  the  length  of  the  longest  boards 
of  equal  length  that  can  be  used  in  fencing  it  without  cutting 
a  board. 

8.  The  driving-  wheels  of  a  locomotive  are  17^-  ft.  in  circum- 
ference, and  the  trucks  10|.     What  distance  must  the  train  move 

o 

to   bring  wheel  and   truck   into   same   relative   positions   as   at 
starting  ? 


138  ARITHMETIC 

9.  The  side  of  a  field  is  19-J-  rd.  in  length,  and  the  end 
IGyt-  rd.  What  is  the  longest  board  that  can  be  nsed  in  fenc- 
ing both  side  and  end  of  the  field,  so  that  no  fraction  (of  a  board) 
may  be  left  ? 

10.  The  distance  between  the  post-office  and   schoolhouse   is 
half  a  mile.     A  carriage  stands  on  the  pavement  in  front  of  the 
schoolhouse.      The  fore  wheels  are  10T45-  ft.  in  circumference,  the 
hind  wheels  15|  ft.      A  chalk  mark  is  made  upon  the  upper 
side  of  each  wheel.     How  often  will  the  4  chalk  marks  be  all  up 
together  while  the  carriage  drives  to  the  post-office  ? 

11.  Show  how  to  find  the  least  common  multiple  of  two  or 
more  fractions.      A,  B,  and  C   start  at  a  given  place  to  travel 
round  an  island  120  mi.   in  circumference ;  A's  rate  is  5J  mi. 
a  da.,  B's  8J,  C's  9-f.      In  what  time  will  they  all  be  together 
again  ? 


CHAPTER   XI 

DECIMALS 

135.  The  standard  unit  of  money  is  $1.  One  dime  is 
one  tenth  of  $  1,  and  1  f  is  one  tenth  of  1  clime.  Hence 
we  write  81,  1  dime,  and  \$  thus,  using  the  dollar  sign: 
11.11. 

Write  in  terms  of  $  1,  the  sum  of  : 

(1)  $1,  2  dimes,  and  4£ 

(2)  1  ten-dollar  bill,  $1,  1  dime,  and  l£ 

(3)  1  one-hundred  dollar  bill,  1  ten-dollar  bill,  $1,1  dime, 
and 


136.  In  the  metric  system  of  measures,  the  standard  unit 
is  1  meter.* 

One  decimeter  is  one  tenth  of  1  meter,  1  centimeter  is 
one  tenth  of  1  decimeter,  and  1  millimeter  is  one  tenth  of 
1  centimeter.  Since  the  same  relation  holds  between  these 
units  as  between  1  dime  and  $1,1^  and  1  dime,  we  can 
express  the  results  of  measurements  in  terms  of  1  meter, 
as  we  express  money  in  terms  of  $  1. 

Hence  if  we  measure  with  the  metric  stick  a  distance  equal 
to  1  meter,  1  decimeter,  and  1  centimeter,  we  can  express  the 
distance  thus:  1.11  meters. 

Measure  the  following  distances  and  express  them  in 
meters  : 

*  See  Chapter  XVIII.  If  the  teacher  has  not  a  metric  stick,  the  work  in 
decimals  may  be  based  on  dollars,  cents,  and  mills. 

139 


140  ARITHMETIC 

(1)  2  meters,  3  decimeters,  and  4  centimeters. 

(2)  1   meter,   2    decimeters,   3    centimeters,   and   5  milli- 
meters. 

(3)  1  meter,  1  decimeter,  1  centimeter,  and  1  millimeter. 

137.  Again,  1    decameter  is    equal  to    10    meters,  and  1 
hectometer  to  10  decameters.     Hence  we  can  write  1  hec- 
tometer,  1   decameter,  1   meter,  1   decimeter,  1  centimeter, 
and  1  millimeter,  thus :  111.111  meters. 

Write  ill  terms  of  the  meter,  the  sum  of : 

(1)  1  hectometer,  2  decameters,  9  meters,   7  decimeters, 
6  centimeters,  and  8  millimeters. 

(2)  3  hectometers,  1  decameter,  5  meters,  2  decimeters,  4 
centimeters,  and  6  millimeters. 

138.  NOTATION  AND  NUMERATION.  -  -  Consider  the  num- 
ber 111 :  the  first  1,  beginning  at  the  right,  denotes  one  unit ; 
the  second,  one  ten  or  ten  units ;  the  third,  one  hundred  or 
one  hundred  units. 

The  third  1  is  equivalent  to  one  hundred  times  the  first  1, 
and  to  ten  times  the  second  1 ;  the  second  is  equivalent  to 
ten  times  the  first  1,  and  to  one  tenth  of  the  third  1 ;  the  first 
1  is  equivalent  to  one  tenth  of  the  second  1,  and  to  one  hun- 
dredth of  the  third  1. 

Now  rewrite  the  number  111,  place  a  point  after  the  first 
1  to  indicate  that  this  1  is  to  be  regarded  as  representing  the 
standard  unit,  and  then  place  after  the  point  three  1's,  so 
that  we  have 

111.111. 

We  may  ask  what  each  of  these  1's  should  mean,  if  the 
same  relation  is  to  hold  among  successive  digits  that  we 


DECIMALS  141 

have  supposed  hitherto  to  hold.  The  1  after  the  point  would 
naturally  mean  one  tenth.  The  next  1  to  the  right  would 
naturally  mean  one  hundredth.  It  is  one  tenth  of  the  pre- 
ceding one  —  that  is,  one  tenth  of  one  tenth. 

Similarly,  the  next  1  would  signify  one  thousandth,  and 
would  equal  one  hundredth  of  the  one  tenth  or  one  tenth 
of  the  one  hundredth.  Thus,  the  number  111.111  may  be 
written  as  follows  :  One  hundred,  one  ten,  one  unit,  one 
tenth,  one  hundredth,  and  one  thousandth. 

Again,  the  1  to  the  extreme  right  is  1  thousandth;  the 
next  1  is,  from  its  position,  equivalent  to  10  thousandths  ; 
and  the  next  1  is  100  thousandths.  So  that  to  the  right 
of  the  point  we  have  111  thousandths.  The  whole  number 
may  now  be  read,  one  hu»dred  and  eleven,  and  one  hundred 
and  eleven  thousandths. 

139.  A  Decimal  Fraction  or  a  decimal  is  one  which  has  for 
its  denominator  10,  100,  1000,  or  some  power  of  10. 

The  Power  of  a  number  is  the  product  found  by  multiply- 
ing the  number  by  itself  one  or  more  times  ;  thus,  100  or  102 
is  the  second  power  of  10  ;  1000  or  103,  the  third  power 
of  10. 

The  denominator  of  a  decimal  fraction  is  never  expressed; 
thus  -f$  and  -ffo  are  written  as  decimals,  .5  and  .57. 

The  point  placed  to  the  right  of  the  one-unit  and  between 
it  and  the  tenth-unit  is  called  the  decimal  point. 

140.  To  change  a  decimal  to  a  vulgar  fraction  in  its  lowest 
terms, 

4O£  _     425     —     85     —  17 
.1^0      -   %-QQ-Q   --   200    -~  TO'' 

6.0364  =  G^fo  =  6  jffo. 
Conversely,  6^ff3ir  =  5.293. 


2  5 


142  ARITHMETIC 

Exercise  94 

1.  Eead  the  numbers  in  Exercises  95  and  96,  expressing  them 
in  terms  of  different  units,  as  1  meter,  1  mi.,  1  Ib. 

2.  Eead  .5,  .05,  .005,  0005. 

3.  What  is  the  ratio  of  .5  to  .05  ?     To  .005  ?     To  .0005  ? 

4.  .0005  is  what  part  of  .005  ?     Of  .05  ?     Of  .5  ? 

5.  Explain  how  it  is  that  the  insertion  of  a  zero  between  the 
point  and  the  5  in  the  decimal  .5  changes  the  value  of  the  deci- 
mal. 

6.  Eead  .5,  .50,  .500,  .5000. 

7.  What  is  the  ratio  of  .5  to  .50  ?     To  .500  ?     To  .5000  ? 

8.  .5000  is  what  part  of  .500  ?     Of  .50  ?     Of  .5  ? 

9.  Explain  why  the  addition  of  zeros  to  the  right  does  not 
change  the  value. 

10.  Name  the  decimal  consisting  of  one  digit  which  lies  near- 
est to  .54,  .66,  .92,  .78,  and  .85. 

11.  Eeduce  the  following  decimals  to  vulgar  fractions  in  their 
lowest  terms : 

.8,  .45,  .06,  .0004,  .375,  .0625,  .006,  .00600. 

12.  Express  the  following  fractions  as  decimals : 

7         43  6  243  29         Q_8         9AQ_  5  4          1  7_      9_ 

TO'    100?    17TO'    10~0"0~J  TOOO'   ^100>  ^u<710000>   J   '  100000' 

Express  in  figures : 

13.  Four  tenths;  two,  and  five  hundredths;  six  thousandths. 

14.  Six  hundred  and  five,  and  twenty-eight-thousandths;  three 
thousand  and  twenty-nine,  and  sixty-five  ten-thousandths ;    two, 
and  four  hundred  and  nine  millionths. 

15.  Eighteen,    and   two   ten-thousandths ;    nine  hundred,  and 
twenty-nine    ten-millionths ;     one    hundred,    and    one    ten-thou- 
sandth. 


DECIMALS  143 

ADDITION  'OF  DECIMALS 
141.   What  is  the  sum  of  4.9,  6.084,  24.32,  and  .8976? 

4  g  In  arranging  the  numbers,  be  careful  to  put  the  deci- 

Q  Qg^  mal  points  directly  under  each  other,  thus  bringing  units 

24  32  under  units,  tenths  under  tenths,  etc.     Then  begin  at 

ggyg  the  lowest  order  and  add  as  if  the  figures  were  integers, 

putting  the  decimal  between  the  unit  and  the  tenths' 

36-2016  place. 

Exercise  95 

Find  the  sum  and  prove  your  answers  correct  by  adding  down 
the  columns : 

1.     53.67  2.       9.7  3.         .8592 

4.009  20.492  913.74 

821.64  .0487  21.0106 

2.182  918.0006  47.9 

4.  Explain  step  by  step  the  process  of  addition  in  the  first 
three  examples. 

Write  in  columns  and  add : 

5.  6.5  +  32.47  +  2.048  +  59. 

6.  .452  +  4.08  +  .646  +  .06  +  49.027. 

7.  4.0406  +  213.939  +  2.91  +  3.04. 

8.  89432.1  +  7.65439  +  .0084  +  8400. 

9.  27.064  +  .0012  +  394.2001  +  .819. 

10.  How  many  yd.  are  there  in  five  pieces  of  cloth,  the  first 
of  which  contains  37.5  yd.,  the  second  26.75,  the  third  14.375, 
the  fourth  36.5,  and  the  fifth  63.125  ? 

11.  Four  sections  of  land  contain  the  following  areas:  24.729 
sq.  mi.,  92.04  sq.  mi.,  8.007  sq.  mi.,  and  36.429  sq.  mi.     Find  the 
total  area. 

12.  Find  the   sum    of    sixty-one  ten-thousandths ;    eight,  and 
seven   thousand   six   hundred   and   ninety-five    ten-thousandths ; 

u 

nine  thousand  seven  hundred  and  eighty-six. 


144  ARITHMETIC 

13.  Find   the    sum   of    the   following :    twenty-four   ten-thou- 
sandths ;    nine,  and   three  thousand  and   four   ten-thousandths ; 
two  hundred  and  seventy-seven,  and  six  hundred  thousandths ; 
nine,  and  nine  thousandths. 

14.  Find  the  sum  of  4  of  the  tenths'  unit,  6  of  the  thousandths' 
unit,  and  8  of  the  millionths'  unit. 

15.  What  is  the  sum  of  4,  6,  9,  7,  5,  8,  3,  6,  and  7,  when  the 
unit  of  value  is  $  1  ?    $.01?    $.001?    $.0001? 

SUBTRACTION  OF  DECIMALS 
142.   From  29.364  take  3.87049. 

29.36400  29.364 

3.87049  3.87049 


As  in  addition  of  decimals,  place  the  decimal  points  under  each  other,  thus 
placing  units  under  units,  tenths  under  tenths,  etc. 

As  the  value  of  the  decimal  is  not  changed  by  annexing  zeros  to  the  right 
of  the  decimal,  annex  in  this  case  two  zeros.  Subtract  as  in  whole  numbers, 
and  place  the  decimal  point  in  the  remainder  between  the  unit  and  the  tenths' 
place. 

Exercise  96 

From        1.    8.43         2.    13.47016         3.    .503  4.    .52 

Take  2.95  2.0984  .28914  .13064 


5.    Explain  step  by  step  the  process  of  subtraction  in  the  first 
three  examples. 

Find  the  difference  and  prove  your  answers  correct : 

6.  .62 -.47.  10.  .07 -.059.  13.  .7304  -  .67. 

7.  .73 -.35.  11.  8.9-3.4265.  14.  4.8295-3.9998. 

8.  .894 -.406.  12.  39.42-15.9879.  15.  2.03  -  .00428. 

9.  .74 -.365. 


DECIMALS  145 

16.  Explain  whether  .067  or  .068  is  nearer  .06748,  and  express 
in  words  the  difference  in  each  case. 

17.  The   length  of   a  seconds   pendulum  is  39.1392  in.,  and 
that    of   a   meter   is   39.371    in.     Find   the   difference    in   their 
lengths. 

18.  Find  the  difference  between  the  length  of   1  meter  and 
lyd. 

19.  From  a  piece  of  cloth  containing  35.5  yd.,  a  merchant 
sold  12.75  yd.     How  much  was  left  ? 

20.  Take  1  millionth  from  1  thousandth. 

21.  Find   the   difference   between   sixty-four   ten-thousandths 
and  seventy-five  tenths.     Add  this  difference  to  their  sum. 

22.  Find  the  difference  between  $  11  -f£$  and  35^. 

23.  The  mercury  in  a  barometer  rose  .121  in.,  .073  in.,  and 
.019  in.  in  three  successive  days ;    it  fell   .054  in.  and  .065  in. 
during  the  two  following  days,  rose  .053  in.  on  the  sixth  day,  and 
fell  .028  in.  011  the  seventh  day.     If  its  height  at  the  beginning 
of  the  first  day  was  30.078  in.,  what  was  its  height  at  the  close 
of  the  seventh  day  ? 

MULTIPLICATION  OF  DECIMALS 

143.    (1)  T7o  multiplied  by  10  =  7,  and  therefore  .7  multiplied  by  10  =  7. 
f §£  multiplied  by  10  =  6T2/,  or  62.7,  and  therefore  6.27  multiplied  by 
10  =  62.7. 

(2)  TV5o  multiplied  by  100  =  75,  and  therefore  .75  multiplied  by  100  =  75. 
ff££  multiplied  by  100  =  &f££  =  627.5,  and  therefore  6.275  multiplied  by 

100  =  627.5. 

(3)  |2i5i   multiplied   by  1000  =  ^2T7o5-  =  6275.8,    and   therefore    6.2758 
multiplied  by  1000  =  6275.8. 

Exercise  97 

1.  Multiply  each  of  the  folloAving  numbers  by  10 : 

.6,  .8,  .84,  .95,  .842,  .763. 

2.  Multiply  by  10 :    .06,  .04,  .005,  .0123. 


146  ARITHMETIC 

3.  Multiply  by  10  :   42.3,  5.69,  .478. 

4.  State  how  to  multiply  a  decimal  by  10,  without  actually 
doing  the  work  of  multiplication. 

5.  Multiply  by  100 :    .84,  9.65,  .763,  .003,  .04,  .246. 

6.  State  how  to  multiply  a  decimal  by  100,  without  actually 
doing  the  work  of  multiplication. 

7.  Multiply  by  1000 :   .982,  .0642,  .0009,  .008,  .0123. 

\  . 

8.  State  how  to  multiply  a  decimal  by  1000.    By  10,000.     By 
100,000. 

9.  Multiply  by  100 :    .86,  8.6,  .9,  .060,  9.8,  .4. 

10.  Multiply  by  1000 :  .594,  5.94,  59.4,  .007,  .07,  .7,  3.14,  2.5. 

11.  State  how  to  divide  a  decimal  by  10.     By  100.     By  1000. 
By  10,000. 

12.  Divide  by  10:  27,  82.19,  4.8,  52.93,  .4,  .06,  .009. 

13.  Divide  by  100:  482,  76,  415.62,  8.1,  .78,  .4,  .09,  789.46. 

14.  Divide  by  1000 :  643,  2459.7,  .69,  2.31,  .03,  .009. 

144.    (1)  Multiply  6.24  by  46. 

6.24 
46 

37.44 

249.6 


287.04 

Here  we  multiply  4  hundredths  by  6,  and  the  product  is  24  huudredths, 
or  2  tenths  and  4  hundredths.  Again,  we  multiply  2  tenths  by  6,  and,  add- 
ing in  the  2  tenths,  the  result  is  14  tenths,  or  1  unit  and  4  tenths,  and  so  on. 
Next,  multiplying  by  4,  we  must  write  the  results  one  place  to  the  left,  as  in 
the  multiplication  of  integers.  In  multiplying,  it  is  as  well  to  omit  the  deci- 
mal points  from  the  partial  products  37.44  and  249.6. 


DECIMALS  14? 

(2)    Multiply  6.24  by  4.6. 

6.24 

4.6 

3744 

2496 

28.704 

This  differs  from  the  former  question  only  in  that  the  multiplier  4.6  is 
one-tenth  of  46,  and  therefore  the  product  is  one-tenth  as  large,  or  28.704. 

145.  From  this  and  other  similar  problems  the  rule  can  be 
deduced : 

To  multiply  two  decimals,  proceed  as  if  they  were  integers, 
and  mark  off  in  the  product  as  many  places  as  there  are  in 
both  multiplier  and  multiplicand. 

146.  (1)  Multiply  2.56  by  .94. 

2.56 
.94 

1024 
2304 
2.4064 

(2)  Multiply  .249  by  .035. 

.249 
.035 
1245 

747 


.008715 
EXPLANATION.  — 

.249  x  .035  =  T2oVo  X  T^o  =  T<rWo5oo  =  -008715. 


Exercise  98 

1.  Find  .05  of  $  26 ;  .06  of  $  248 ;  .04  of  $  16. 

2.  Find  .08  of  $46.50;  .03  of  $894.75;  .07  of  $2389.20. 

3.  Find  .3  of  $  250 ;  .9  of  $  64 ;  .8  of  $  76. 

4.  Find  .025  of  $  324;  .035  of  $  704.60;  .045  of  $  852.94. 


148  ARITHMETIC 

5.  Find  .375  of  45  mi. ;  .0625  of  640  A. ;  .0875  of  $415.60. 

Multiply : 

6.  4.8  x5.12;  .21x4.67. 

7.  3.1416  x  .02;  1.46  x  .39. 

8.  .004  x.99;  .004  x  .005. 

9.  $  249  x  1.04. 

10.  .84  x  .251 ;  2.04  x  .0037. 

11.  .8  x  .8;  .09  x  .09. 

12.  .1  x  .1;  .01  x  .01. 

13.  .2  x  .2;  .7  X  .7. 

14.  .375  x  2.15 ;  .0375  x  2.15. 

15.  .051  x  .042 ;  .014  x  .0038. 

16.  Find  the  area  of  a  rectangle  6.2  ft.  long  by  4.7  ft.  wide. 

17.  Find  the  area  of  a  rectangle  7.5  ft.  wide  by  3.4  ft.  long. 

18.  Find  the  value  of  a  rectangular  solid  whose  dimensions 
are  2.5,  4.6,  and  5.8  in. 

19.  Find  the  volume  of  a  solid  whose  dimensions  are  8.8,  6.6, 
and  4.4  in. 

20.  If  my  gain  on  selling  an  article  is  measured  by  the  num- 
ber .23  and  the  cost  $  265,  find  the  gain. 

21.  If  my  loss  on  selling  an  article  that  cost  $  226.50  is  meas- 
ured by  the  number  .34  and  the  cost,  find  the  loss. 

22.  Multiply  $  240  by  1.05  twice  in  succession. 

23.  Multiply  $  325  by  1.06  three  times  in  succession. 

24.  Multiply  f  415.80  by  1.04  twice  and  the  result  by  1.02. 

25.  Multiply  $  75  by  1.045  three  times  in  succession. 

26.  Multiply  $  975  by  1.065  three  times  in  succession. 

27.  Multiply  3.1416  by  8  ;  by  7.5 ;  by  .04. 

28.  Multiply  7.48  by  9.1 ;  by  .04  ;  by  .006. 

29.  Multiply  the  square  of  5  by  3.1416. 


DECIMALS  149 

30.  Measure  in  inches  and  decimals  of  an  inch  the  diameters 
of  several  circles  and  multiply  the  result  in  each  case  by  3.1416. 
Then  measure  the  circumference  and  note  which  are  greater,  the 
products  or  the  lengths  of  the  circumferences. 

Exercise  99 

1.  The   length   of   a  wall,   according  to  the   French   metric 
system,  is  9.48  meters.     Find  its  length  in  in.,  the  length  of  1 
meter  being  39.371  in. 

2.  Multiply  the  sum  of  2.616,  .00132,  and  1.0448  by  .62639. 

3.  A  piece  of  land  is  63.5  rd.  long,  and  27.75  rd.  wide.    What 
will  it  cost  to  fence  it  at  $  .875  per  rd.  ? 

4.  Multiply  10.5  by  1.05  and  reduce  the  result  to  a  fraction 
in  its  lowest  terms. 

5.  The   specific   gravity  of   atmospheric  air  compared  with 
water  is  .0012.     I  ask  for  the  specific  gravity  of  common  gas,  and 
am  told  it  is  .45  compared  with  air.     Find  its  specific  gravity 
compared  with  water. 

6.  A    lumber    merchant    bought    106,250    ft.    of   lumber   at 
$  14.375  per  M.,  and  retailed  it  at  $  1.75  per  C.     Find  his  gain. 

7.  Water  is  composed  of  two  gases,  oxygen  and  hydrogen,  in 
the  proportion  of  88.9  to  11.1.     What  weight  is  there  of  each  in 
a  cubic  yard  of  water  ?     (1  cu.  ft.  of  water  weighs  1000  oz.) 

8.  The  chain  for  measuring  land  is   66  ft.  long.     What  is 
the  length  in  yards  of  a  fence  that  measures  2456  ch.,  and  how 
much  would  it  cost  at  $  8.86  per  yd.  ? 

9.  Find  the  area  of   a   rectangular   field  whose   breadth  is 
78.23  ch.,  and  length  85.40  ch.,  there  being  10  sq.  ch.  in  1  A. 

10.  A  farmer  bought  48.125  T.  of  hay ;  for  20.25  T.  of  it  he 
paid  $16  per  T.,  and  for  the  rest  $18.2625  per  T. ;  he  sold 
the  whole  at  the  average  price  of  $  .945  per  cwt.  How  much 
did  he  gain  or  lose  ? 


150  ARITHMETIC 

11.  A  person  sold  .15  of  an  estate  to  one  person,  and  then  TgT 
of  the  remainder  to  another  person.     What  part  of  the  estate  did 
he  still  retain  ? 

12.  If  a  business  produces  an  annual  return  of  $  12,000,  and  of 
three  partners  one  has  .465  and  another  .28  share  of  the  profits, 
how  much  money  falls  to  the  share  of  the  third  partner  ? 

13.  A  merchant  sells  28.5  yd.  of  cloth  which  cost  him  25^ 
a  yd.,  for  37.5  ^  a  yd.     What  was  his  gain  ? 

DIVISION  OF  DECIMALS 

8)24  8)2.4  8). 24 

3  .3  .03 

That  is,  24  divided  by  8  =  3. 

24  tenths  divided  by  8  =  3  tenths  =  .3. 

24  hundredths  divided  by  8  =  3  hundredths  =  .03. 

6). 0018  46)251.62(5.47 

.0003  230 

216 
184 
322 
322 

That  is,  18  ten-thousandths  -*-  6  =  3  ten-thousandths  =  .0003  ;  and  25,162 
hundredths  -4-  46  =  547  hundredths  =  5.47. 

147.  5)15  50)150  500)1500  5000)15000 

333  3 

Therefore,  if  we  multiply  the  divisor  by  10, 100,  1000,  and 
so  on,  and  at  the  same  time  multiply  the  dividend  by  10, 
100,  or  1000,  and  so  on,  the  quotient  remains  unchanged. 

(1)  Find  the  value  of  4.1262  -*-  .69. 

The  quotient  of  4.1262  -5-  .69  is  the  same  as  that  of  412.62  -s-  69.  Here 
we  multiplied  each  number  by  100. 


DECIMALS  151 

69)412.62(5.98 
345 
676 
621 
552 
552 

/.  4.1262  --  .69  =  5.98. 

In  this  division  the  412  is  412  units,  and  the  quotient  5  is  therefore  5  units ; 
the  first  remainder,  676,  is  676  tenths,  and  the  quotient  9  is  therefore  9  tenths  ; 
the  second  remainder,  552,  is  552  hundredths,  and  the  quotient  8  is  therefore 
8  hundredths. 

(2)  Find  correct  to  the  third  decimal  place  the  quotient 
of  8.94  -v-  3.1416. 

3.1416)8.94( 
31416)89400(2.845 

62832 

265680 

251328 


143520 
125664 

17856 

15708 
2148 

.-.  8.94  -i-  3.1416  =  2.845,  correct  to  three  decimal  places. 

Exercise  100 

Divide,  proving  your  answer  correct  to  every  third  question : 

1.  25.68-3.21.  3.    8.54  -.07. 

2.  10.836 -s- 5.16.  4.    $49.92 -.065. 

5.  $  54.75  —  .98,  correct  to  three  decimal  places. 

6.  $  75.60  —  .99,  correct  to  three  decimal  places. 

7.  $64.26-1.02. 

8.  $  84.3648  —  1.04  twice  in  succession  and  the  result  by  1.02. 


152  ARITHMETIC 

9.    $  16989.7728  -=~  1.04  twice  in  succession  and  the  result  by 
1.02. 

10.  2450.90  -T-  .998,  correct  to  three  decimal  places. 

11.  $23SSl-f-1.06  three  times  in  succession,  correct  to  three 
decimal  places. 

12.  $11.19195  -=-$4.8665. 

13.  .00081  ~  27,  and  1.77089  by  4.735. 

14.  1  -  .1,  by  .01,  and  1  -  .0001. 

15.  31.5 -.126;  5.2-V-.32. 

16.  12.6  -  .0012,  and  .065341  -  .000475. 

17.  3.012  -  .0006. 

18.  130.4  -  .0004  and  .004,  and  46.634205  -  4807.65. 

19.  1.69  -T- 1.3,  by  .13,  by  13,  and  by  .013. 

20.  816 -.0004. 

21.  .00005  -  2.5,  by  25,  and  by  .0000025. 

22.  32.5  -7-  8.7 ;  .02  -7- 1.7,  correct  to  four  decimal  places. 

23.  .009384  -7-  .0063,  correct  to  four  decimal  places. 

24.  37.24  -7-  2.9  ;  .0719  —  27.53,  correct  to  four  decimal  places. 

25.  Measure  in  inches  and  decimals  of  an  inch  the  diameter  of 
any  circle,  and  divide  the  result  into  the  length  of  the  circum- 
ference.    Do  this  until  you  find  the  quotient  to  be  nearly  3.1416. 

Exercise  101 

1.  If  a  gal.  contains  231  cu.  in.  and  a  cu.  ft.  1728  cu.  in., 
find  correct  to  two  decimal  places  the  number  of  gal.  in  a  cu.  ft. 
of  water. 

2.  Using  the  result  obtained  in  question  1,  find  the  number 
of  gal.  of  water  that  a  rectangular  tin  2  ft.,  by  3  ft.,  by  4  ft. 
will  hold. 

3.  If  a  bu.  contain  2150.42  cu.  in.,  find  the  number  of  cu.  in. 
in  a  dry  qt. 


DECIMALS  153 

4.  Find  correct  to  three  decimal  places  the  number  of  cu.  ft. 
in  a  bu. 

5.  By  what  decimal  part  of  1  in.  does  .0009  of  1  ft.  exceed 
.00003  of  1  yd.  ? 

6.  Find  cost  of  7225  Ib.  coal  at  $  7.25  per  ton  of  2000  Ib. 

7.  Find  the  value  of 

3.0005  x  .006 
.0009 

8.  A  man  paid  $2,896,863.50  for  land  and  sold  56.25  A.  at 
$  31  an  A. ;  the  remainder  then  stood  him  at  $  20.05  an  A.     How 
many  A.  did  he  buy  ? 

9.  Water  expands  when  freezing  so  that  a  cu.  ft.  of  water 
becomes  1.089  cu.  ft.  of  ice.     Find  how  many  cu.   ft.   of  water 
there  are  in  an  iceberg  which  is  900  ft.  long,  88  ft.  broad,  and 
220  ft.  higii. 

10.  Show  by  examples  that  a  decimal  is  divided  by  10,000  by 
removing  the  decimal  point  in  the  dividend  four  places  towards 
the  left. 

11.  A  creditor  receives  $  1.50  for  every  $4  of  what  was  due  to 
him,  and  thereby  loses  $  301.05.     What  was  the  sum  due  ? 

12.  Divide  .0075  by  6.4,  and  explain  the  reason  for  fixing  the 
position  of  the  decimal  point  in  the  quotient. 

13.  A  person  expended  $  55.92  in  tea  at  $.875  per  Ib.,  coffee 
at  $.1875,  and  sugar  at  $.1025,  buying  an  equal  quantity  of  each. 
How  many  Ib.  of  each  did  he  buy  ? 

14.  A  gentleman  whose  real  property  is  .834  of  his  personal 
property  leaves  the  former,  amounting  to  $  10,008,  to  his  eldest 
son ;  and  the  latter  to  be  equally  shared  by  him  and  two  others. 
Find  the  amount  received  by  each. 

15.  A  merchant  expended  $  280.60  in  purchasing  cloth  at  95^ 
a  yd.,  at  $  1.37  a  yd.,  and  at  73^  a  yd.,  buying  the  same  quantity 
of  each.     Find  the  entire  number  of  yd.  purchased. 


154  ARITHMETIC 

16.  Suppose  unity  to  represent  .0012,  what  number  represents 
.0001  ? 

17.  Find  the  earth's  equatorial  diameter  in  miles,  supposing 
the  sun's  diameter,  which  is  111.454  times  as  great  as  the  equa- 
torial diameter  of  the  earth,  to  be  883,345  mi. 

18.  The  French  meter  is  39.371  in.  in  length.      Express  the 
length  of  25  meters  as  a  decimal  of  an  English  mi.,  there  being 
5280  ft.  in  1  mi.  and  12  in.  in  1  ft. 

19.  The  total  Indian  population  on  the  reservations  in  1880 
was  255,327,  while  the  area  of  the  Indian  reservations,  in  the 
United  States,  was  241,800  sq.  mi.     What  average  quantity  of 
land  was  occupied  by  100  Indians  ? 

20.  The  total  Indian  population  on  the  reservations  in  1893 
was   249,366,    while   the   area   of   the   Indian   reservations   was 
134,176  sq.   mi.     What  average  quantity  of  land  was  occupied 
by  100  Indians  ? 

21.  The  area  of  the  reservations  in.  1893  was  how  many  thou- 
sandths of  that  in  1880  ? 

22.  The   Indian   population  on  the  reservations  in  1893  was 
how  many  thousandths  of  that  in  1880  ? 

23.  Find  the  quantity  of  coal  consumed  by  a  steamer  for  a 
voyage  of  4043  mi.,  supposing  her  rate  per  hr.  to  be  16.172  mi. 
and  her  consumption  of  coal  87  T.  per  da. 

REDUCTION  OF  DECIMALS 

148.    (1)  Reduce  .275  to  a  common  fraction. 

•275  = 


(2)  Reduce  to  a  common  fraction  .08^. 


-         -. 

100      300      12 


DECIMALS  155 

Exercise  102 

Reduce  to  common  fractions  in  their  lowest  terms : 

1.  .5.  6.    .625.  11.    .331  16.  8.9375. 

2.  .25.  7.    .125.  12.    .031  17.  29.975. 

3.  .75.  8.    .0625.  13.    .061.  18.  18.06|. 

4.  .60.  9.    .875.  14.    .66|.  19.  6.004. 

o  O 

5.  .375.  10.    .16|.  15.    .14f  20.    249.00075. 

21.  If  my  gain  on  selling  a  Ib.   of  tea  is  measured  by  the 
number  .125,  and  the  cost  72^  what  was  my  gain  on  1  Ib.  ? 

22.  If  a  crate  of  berries  which  cost  $  1.35  was  sold  at  a  gain 
of  .22-2-  of  the  cost,  find  the  gain. 

23.  I  bought  a  farm  for  $4800,  and  sold  it  at  a  loss  of  .375  of 
the  cost  price.     Find  the  selling  price. 

24.  A  merchant  sold  coffee  at  a  gain  of  .33^  of  the  cost.     If 
his  gain  on  a  quantity  of  coffee  was  $  12,  what  did  it  cost  him  ? 

25.  A  grain  merchant  sold  wheat  for  $  3400,  gaining  .061  Of 
the  cost.     Find  the  cost  price. 

149.    (1)  Divide  7  by  8,  expressing  the  result  as  a  decimal. 

8)7.000 
.875 

Now  7-8  =  |.     t.m  |  _  > 

PROOF 

•875  =  tffs  =  m  =  f 

(2)  Reduce  -J-|-  to  a  decimal. 

16)150(.9375 
144 

60 
48 

120 
112 

80 
,-.  if  =.9375.  80 


156  ARITHMETIC 


(3)  Reduce  9^|  to  a  decimal  correct  to  four  decimal  places. 


23)180(.7826 
161 
190 
184 
60 
46 


140 
138 

2 

.-.  Off  =  9.7826,  correct  to  four  decimal  places. 

Exercise  103 

Reduce  to  decimals  and  prove  results  : 

1         1       1       3  01357  Q         1          5         9  /I          9        1  9 

-1*      ¥>     2>     4'  *"    8"?     8"?    ~8>    "8*  Om     T6"?    T"6?    T6'  *•     "5T?    2"0' 

5-    T6268>  fV45?  rio-  6.    611    3if.  7.    f,  |,  }£,  fj. 

8.    Find  correct  to  four  decimal  places  the  value  of  T4^,  1^, 

14       16 
"        "21' 


9.    Butter   bought    for   25^   a   Ib.    was    sold    for    28^    a   Ib. 
Express  the  gain  as  a  decimal  of  the  cost. 

10.  I  bought  a  store  for  $  6912,  and  sold  it  for  $  5184.     Ex- 
press the  selling  price  as  a  decimal  of  the  cost. 

11.  A  real  estate  agent  sold  land  which  cost  him  $6240  at 
a  loss  of  $  2340.     What  was  his  loss  on  each  $  1000  invested  ? 

12.  A  ch.  contains  66  ft.,  and  a  mi.  5280  ft.     What  decimal 
part  of  a  mi.  is  a  ch.  ? 


CHAPTER   XII 

COMPOUND  QUANTITIES 

150.  Quantities  like  4  yd.,  3^  lb.,  and  6J  gal.  are  called 
simple  quantities,  because  they  are  expressed  in  terms  of  a 
single  unit  of  measurement. 

Quantities  like  3  lb.  8  oz.,  6  gal.  1  qt.,  are  called  compound 
quantities,  because  they  are  expressed  in  terms  of  two  or  more 
units  of  measurement. 

151.  The  units  of  money  are  the  units  which  are  used  to 
measure  the  values  of  things.    The  one-dollar  gold  piece  is  at 
present  (July,  1896)  the  prime  unit  or  standard  of  value  in 
the  United  States  and  Canada. 

152.  UNITS  OF  VALUE 

UNITED   STATES  MONEY 

10  mills  (m.)  =  1  cent  (ct.  or  ^) 
10  cents          =  1  dime  (d.) 
10  dimes         =  1  dollar  ($) 
10  dollars       =  1  eagle  (E.) 

The  coins  of  the  United  States  are : 

Bronze :  the  cent. 

Nickel :  the  five-cent  piece. 

Silver :  the  dime,  quarter-dollar,  half-dollar,  and  dollar. 

Gold  :  the  quarter-eagle,  half-eagle,  eagle,  and  double-eagle. 

153.  Sterling  Money  is   the   money  of  Great   Britain   and 
Ireland. 

157 


158  ARITHMETIC 

The  prime  unit  is  1  pound,  whose  value  is  $  4.8665. 
The  pound,  when  coined,  is  called  the  sovereign. 

BRITISH  OR  STERLING  MONEY 

4  farthings  (far.)  =  1  penny  (d) 
12  pence  =  1  shilling  (s.) 

20  shillings  =  1  pound  (£) 

5  shillings  =  1  crown 

21  shillings  =  1  guinea 

154.    The  unit  of  French  Money  is  1  franc,  which  is  worth  19.3^. 
The  unit  of  German  Money  is  1  mark,  which  is  worth  23.85^. 


Exercise  104 

1.  How  many  mills  are  there  in  2^  ?    3^  ?    \<t  ?     \\t  ?    2 

2.  How  many  cents  are  there  in  40  mills  ?     60  mills  ?     15 
mills?     5  mills?     25  mills  ? 

3.  State  orally  the  table  of  English  Money. 

4.  Reduce  to  farthings  :  3d. ;  6d.  2  far. ;  9d.  3  far. 

5.  Reduce  to  pence:  4s.;  8s.  5d  ;  12s.  6 d. 

6.  Reduce  to  shillings :  £  7  ;  £  2  12s. ;  £  9  7s. 

7.  How  many  pence  are  there  in  |-s.  ?    |s.  ?    |s.  ?   -f  s.  ?    -f  s.  ? 

8.  How  many  shillings  and  pence  are  there  in  £  -|  ?     £  ^  ? 

£5.9       £1.9 
•^  6   '         ^  8   * 

9.  How  many   shillings  are  there   in    £  .3  ?     £  .7  ?     £  .25  ? 
£  .331  ? 

10.  What  fraction  of  a  shilling  is  3d?     4d.  ?     Sd.  ?     9cl? 
10  d.? 

11.  What  is  the  value  of  £  1  in  American  money?  Of  £  10  ? 
Of  £100? 

12.  How  many  shillings  and  pence  are  there  in  60  d.  ?    84  d,  ? 
:;'.)/.?    58 cZ.?    112 d.  ? 


COMPOUND   QUANTITIES  159 

13.  How  many  pounds  and  shillings  are  there  in  80s.  ?    65s.  ? 
120s.?    48s.? 

14.  How  many  pounds  and  shillings  in  1  guinea  ?    4  guineas  ? 
6  guineas  ?    9  guineas  ? 

15.  What  decimal  of  a  pound  is  10s.  ?    12s.?    17s.?   24s.? 

16.  What  part  of  a  crown  is  1  s.  ?     How  many  crowns  in  10s.  ? 
20s.? 

17.  Express  2  guineas  in  sovereigns  and  shillings. 

18.  What  is  the  value  of  10  francs  in  U.  S.  money  ? 

19.  What  is  the  value  of  100  marks  in  U.  S.  money  ? 

20.  What  is  the  cost  in  cents  of  3  books  at  1  franc  each  ? 

21.  What  is  the  difference  in  value  between  100  marks  and 
100  francs  ? 

UNITS  OF  WEIGHT 

155.   Avoirdupois  Weight  is  used  for  weighing  everything  ex- 
cept jewels,  precious  metals,  and  medicines  when  dispensed. 
The  prime  unit  of  weight  is  1  pound  Avoirdupois. 

AVOIRDUPOIS  WEIGHT 

16  ounces  (oz.)      =  1  pound  (Ib.) 
100  pounds  =  1  hundredweight  (cwt.) 

20  hundredweight  =  1  ton  (T. ) 

In  the  United  States  Custom  House,  and  in  weighing  iron  and  coal  at  the 
mines,  the  long  hundredweight  and  the  long  ton  are  used. 

112  pounds  =:  1  long  hundredweight 
2240  pounds  =  1  long  ton 

One  pound  Avoirdupois  —  7000  grains 
One  ounce  Avoirdupois  =  437  J-  grains 

Exercise  105 

1.  State  orally  the  table  of  Avoirdupois  Weight, 

2.  Eeduce  to  oz.  :  1  Ib.  8  oz. ;  2  Ib.  4  oz., 


160  ARITHMETIC 

3.  Express  1  Ib.  8  oz.  as  a  fraction  of  2  Ib.  4  oz. 

4.  Express  1  Ib.  8  oz.  as  a  decimal  of  2  Ib.  4  oz. 

5.  What  part  of  1  Ib.  is  4  oz.  ?     12  oz.  ?     2   oz.  ?     8  oz.  ? 
Express  your  results  also  as  decimals. 

6.  What  part  of  1  T.  is  400  Ib.  ?    800  Ib.  ?    1500  Ib.  ? 

7.  A  coal  dealer  buys  coal  by  the  car-load  at  the  mines.    How 
many  more  Ib.  of  coal  does  he  get  for  1  T.  than  he  gives  ? 

8.  Show  by  dividing  7000  gr.  by  16,  that  1  oz.  Avoir,  is  equal 
to  4371  gr. 

9.  One  oz.  is  what  part  of  1  Ib.  ?     •£  of  an  oz.  is  what  part  of 
lib.? 

10.    A  farmer  sells  3  cows  whose  united  weight  is  1  T.  5  cwt. 
What  is  the  average  weight  of  the  cows  ? 

156.    Troy  Weight  is  chiefly  used  for  weighing  gold,  silver, 
and  jewels. 

TROY  WEIGHT 

24  grains  (gr.)      =  I  pennyweight  (pwt.) 
20  pennyweights  =  1  ounce  (oz.) 
12  ounces  =  1  pound    (Ib.) 

One  pound  Troy  =  5760  grains 
One  ounce  Troy  =    480  grains 

A 

* 

Exercise  106 

1.  State  orally  the  table  of  Troy  Weight. 

2.  Reduce  to  gr. :  1  pwt.  16  gr. ;  2  pwt.  12  gr. 

3.  What  is  the  ratio  of  2  pwt.  12  gr.  to  1  pwt.  16  gr.  ? 

4.  Express  1  pwt.  16  gr.  as  a  decimal  of  2  pwt.  12  gr. 

5.  Reduce  1  oz.  to  gr. 

6.  Divide  5760  gr.  by  12,  and  verify  your  result  in  question  5. 


COMPOUND   QUANTITIES  161 

7.  By  liow  many  gr.  is  1  Ib.  Avoir,  heavier  than  1  Ib.  Troy? 

8.  By  how  many  gr.  is  1  oz.  Troy  heavier  than  1  oz.  Avoir.  ? 

9.  How  many  oz.  and  pwt.  are  there  in  J  Ib.  ?     f  Ib.  ?     f  Ib.  ? 


lib.? 


10.  How  many  pwt.  are  there  in  .25  oz.  ?     .4  oz.  ?     .35  oz.  ? 

11.  What  part  of  a  Ib.  is  8  oz.  ?    1  oz.  ?    |  oz.  ?    f  Oz.  ? 

12.  If  coal   is  worth  $6  a  T.,  how  many  Ib.  can  be   bought 


for  $3?    $2? 

13.  A  cu.  ft.  of  water  contains  1000  oz.  How  many  Ib.  does  a 
cu.  ft.  of  water  weigh  ? 

157.  Druggists  buy  their  medicines  by  Avoirdupois  Weight, 
but  use  Apothecaries'  Weight  in  mixing  and  in  selling  medi- 
cines. 

APOTHECARIES'   WEIGHT 

20  grains  (gr.)=  1  scruple  (3) 

3  scruples        =  1  dram  (3) 

8  drams  =  1  ounce  (  ^  ) 

12  ounces          =  1  pound  (Ib.) 

One  pound  Apothecaries'  weight  =  5760  grains 
One  ounce  Apothecaries'  weight  =    480  grains 

Exercise  107 

1.  State  orally  the  table  of  Apothecaries'  Weight. 

2.  How  many  ounces  in  16  3  ?    40  3  ?    72  3  ? 

3.  How  many  drams  in  9  3  ?    18  3  ?    54  3  ? 

4.  What  part  of  a  pound  is  4  5  ?    9  5  ?    10  3  ? 

5.  How  many  scruples  in  33  and  23?     In  40  gr.?     In  120 
gr.?    . 

M 


ARITHMETIC 

UNITS  OF  LENGTH 

158.  The  prime  or  standard  unit  of  length  is  1  yard. 

LONG  MEASURE 

12  inches  (in.)  =  1  foot  (ft.) 

3  feet  =  1  yard  (yd.) 

5J  yards  or  16i  feet  =  1  rod  (rd.) 
320  rods  =  1  mile  (mi.) 

1  mi.  =  320  rd.  =  1760  yd.  =  5280  ft. 

A  hand,  used  in  measuring  the  height  of  horses,  =  4  in.  ;  a  knot,  used  in 
navigation,  =  6086  ft.  or  1.15  mi. 

A  fathom,  used  in  measuring  depth  at  sea,  =  6  ft. 

159.  Surveyor's    Linear    Measure   is    used   by  surveyors  in 
measuring  land.    The  prime  unit  is  1  chain,  called  Grunter's 

Chain. 

SURVEYOR'S  LINEAR  MEASURE 

100  links  (!.)=!  chain  (ch.) 
80  chains      =  1  mile  (mi.) 

1  ch.  =  4  rd.  =  22  yd.  =  66  ft.  =  792  in. 
1  link  =  7.92  in. 

160.  Mark  off  in  the  schoolroom  1  ft.,  1  yd.,  and  1  rd. 
Locate  two  points  exactly  1  mi.  apart. 

Exercise  108 

1.  How  many  in.  are  there  in  1  yd.  ?     \  yd.  ?     f  yd.  ?     J-  yd.  ? 
£  vd  ? 

6    J         ' 

2.  Eeduce  to  yd.  :  4  rd. ;  8  rd. ;  32  rd. ;  320  rd. ;  1  mi. 

3.  Reduce  to  ft.  :  18  yd. ;  76  yd.  5  176  yd. ;  1760  yd. ;  1  mi. 

4.  Show  that  1  mi.  =  320  rd.  =  1760  yd.  -  5280  ft.  =  63,360  in. 

5.  How  many  yd.   are  there  in   .3  mi.  ?      .8  mi.  ?     .25  mi.  ? 
.061  mi.  ? 

6.  What  part  of  a  ft.  is  3  in.  ?  6  in.  ?  8  in.  ?  10  in.  ? 

7.  What  part  of  a  yd.  is  12  in.  ?  18  in.  ?  24  in.  ?  27  in.  ? 


COMPOUND   QUANTITIES  163 

8.  What  part  of  a  rcl.  is  1  yd.  ?  3  yd.  ?  5  yd.  ?  51  yd.  ? 

9.  What  part  of  a  mi.  is  40  rd.  ?  200  yd.  ?  280  yd.  ? 

10.  How  many  yd.  are  there  in  1  rd.  ?     How  many  ft.  ?    How 
many  in.  ? 

11.  How  many  ch.  are  there  in  320  rd.  ?     How  many  rd.  are 
there  in  1  ch.  ? 

12.  Show  that  1  ch.  =  4  rd.  =  22  yd.  =  66  ft.  =  792  in. 

13.  How  many  in.  are  there  in  100  links  ?     In  1  link  ? 

UNITS  OF  SURFACE  OR  SQUARE  MEASURE 

161.  Surface  has  two  dimensions,  —  length  and  breadth. 

162.  The  prime  unit  of  area  is  1  square  yard,  which,  like 
1  square  inch,  1  square  foot,  1  square  rod,  and  1  square  mile, 
is  derived  from  the  corresponding  unit  of  linear  measure. 

The  measure  of  1  ft.  is  12,  the  unit  being  1  in. 
The  measure  of  1  sq.  ft.  is  144,  the  unit  being  1  sq.  in. 
.-.   1  sq.  ft.  =  144  sq.  in. 

The  measure  of  1  yd.  is  3,  the  unit  being  1  ft. 
The  measure  of  1  sq.  yd.  is  9,  the  unit  being  1  sq.  ft. 
.-.  1  sq.  yd.  =  9  sq.  ft. 

The  measure  of  1  rd.  is  5|,  the  unit  being  1  yd. 

The  measure  of  1  sq.  rd.  is  (5£  x  5i),  or  30|,  the  unit  being  1  sq.  yd. 
.-.  1  sq.  rd.  =  30£  sq.  yd. 

Illustrate  the  above  by  drawing  1  sq.  ft.,  1  sq.  yd.,  and  1  sq  rd.,  and  divid- 
ing each  into  the  next  lower  units  of  area. 

In  the  case  of  1  sq.  rd.  draw  according  to  the  scale  of  4  in.  to  1  rd. 

SURFACE  OR  SQUARE  MEASURE 

144  square  inches  (sq.  in.)=  1  square  foot  (sq.  ft.) 
9  square  feet  =  1  square  yard  (sq.  yd.) 

30|  square  yards  =  1  square  rod  (sq.  rd.) 

160  square  rods  =  1  acre  (A.) 

640  acres  =  1  square  mile  (sq.  mi.) 

10  square  chains  =  1  acre  ;  1  acre  =  4840  square  yards 


164 


ARITHMETIC 


163.  A  township  is  6  mi.  square,  and  is  divided,  as 
in  the  accompanying  figure,  into  36  sections,  each  1  mi. 
square. 


6 

5! 

4 

3 

2 

1 

7 

8 

9 

10 

11 

12 

18 

17 

16 

15 

14 

13 

19 

20 

21 

22 

23 

24 

30 

29 

28 

27 

26 

25 

31 

32 

33 

34 

35 

36 

N   E.  U 

160  ACRES 

w.  y. 

320  ACRES 

N.  J£OF  S.E.  X 
80  ACRES 

S.W.&OF  S.  E.  X  OF 
S.E.    H         S.  E.   M 
40  ACRES    40  ACRES 

TOWNSHIP. 


SECTION. 


Locate  sections  8,  22,  and  36  in  the  drawing.  Draw  a 
township  on  a  scale  of  1  in.  to  1  mi.,  and  divide  it  into 
36  sections,  numbering  each  section.  Divide  one  of  these 
sections  into  4  square  farms  of  160  A.  each,  and  name 
each  farm  according  to  its  position  in  the  section.  Divide 
a  second  section  into  8  rectangular  farms  of  80  A.,  and  a 
third  into  16  square  farms  of  40  A.,  and  locate  each  farm  as 
before. 

Exercise  109 

1.  How  many  sq.  in.  are  there  in  2  sq.  ft.  ?   4  sq.  ft.  ?    9  sq.  ft.  ? 

2.  How  many  sq.  ft.  in  5  sq.  yd.  ?  -J-  sq.  yd.  ?  \  sq.  yd.? 

3.  One  sq.  rd.  is  equal  to  how  many  sq.  yd.  ?    4  sq.  rd.  ?  16 
sq.  rd.  ?  160  sq.  rd.  ?  1  A.  ? 

4.  What  part  of  a  sq.  rd.  is  1  sq.  yd.  ? 

5.  Keduce  to  sq.  rd.  :  484  sq.  yd.;  1511  sq.  yd. 

6.  What  part  of  an  A.  is  80  sq.  rd.  ?  120  sq.  rd.  ? 

7.  How  many  sq.  ch.  are  there  in  160  sq.  rd.  ?     One  sq.  ch. 
equals  how  many  sq.  rd.  ? 


COMPOUND   QUANTITIES  165 

8.  Reduce  to  A.  :  5  sq.  mi. ;  8  sq.  mi. 

9.  If  1  sq.  mi.  is  the  unit  of  area,  find  the  number  which 
expresses  the  measure  of  2560  A.     Of  4  townships. 

10.  Into  how  many  townships  can  a  county  be  divided  which 
contains  324  sq.  mi.  ? 

11.  What  is  the  area  of  a  square  6  ft.  in  length  ? 

12.  What  is  the  difference  in  area  between  two  figures,  one  6 
in.  sq.  and  the  other  containing  6  sq.  in.  ? 

Illustrate  by  a  drawing. 

UNITS  OF  VOLUME 

164.  A  volume  has  three   dimensions,  —  length,  breadth, 
and  thickness. 

165.  The  prime   unit  of  volume  is  1   cubic  yard,  which, 
like  1  cubic  inch  and  1  cubic  foot,  is  derived  from  the  cor- 
responding unit  of  linear  measure. 

The  measure  of  ttie  volume  of  1  cu.  ft.  =  12  x  12  x  12  =  1728,  the  unit 
of  volume  being  1  cu.  in. 

The  measure  of  the  volume  of  1  cu.  yd.  =  3  x  3  x  3  =  27,  the  unit  of 
volume  being  1  cu.  ft. 

CUBIC  OR  VOLUME  MEASURE 

1728  cubic  inches  (cu.  in.)=  1  cubic  foot  (cu.  ft.) 
27  cubic  feet  =  1  cubic  yard  (cu.  yd.) 

166.  Fire  wood  and  rough  stone  are  measured  by  the  cord  (cd.).     The 
cord  is  a  pile  8  ft.  long,  4  ft.  wide,  and  4  ft.  high.     It  contains  128  cu.  ft. 
One  cord  foot  (cd.  ft.)  is  1  ft.  in  length  of  the  cord.     Its  volume  is  16  cu.  ft. 

A  cu.  yd.  of  earth  is  called  a  load. 

How  many  loads  of  dirt  are  there  in  a  pile  15  ft.  long,  12  ft.  wide,  and 
0  ft.  deep  ? 

167.  Mark  off  in  one  corner  1  cu.  ft.,  1  cu.  yd.,  and  1  cd. 
Divide  the  cd.  into  cd.  ft. 


166  ARITHMETIC 

UNITS  OP  CAPACITY 

168.  The  prime  unit  of  capacity  is  1  gallon. 

LIQUID  MEASURE 

4  gills  (gi.)=  1  Pint  (pt.) 
2  pints         =  1  quart  (qt.) 
4  quarts      =  1  gallon  (gal.) 

169.  The  capacity  of  cisterns,  reservoirs,  and  the  like  is  often  expressed 
in  barrels  (bbl.)  of  311  gal.  each,  or  in  hogsheads  (hhd.)  of  63  gal.  each.    A 
gal.  contains  231  cu.  in.     Have  a  tin  box  made  11  in.  long,  7  in.  wide,  and 
3  in.  deep,  and  note  that  1  gal.  of  water  will  just  fill  it. 

DRY  MEASURE 

2  pints  (pt. )  =  1  quart  (qt. ) 
8  quarts         =  1  peck  (pk.) 
4  pecks          =  1  bushel  (bu. ) 
One  bushel  contains  2150.42  cubic  inches 

170.  Apothecaries'  Fluid   Measure  is  used  by  druggists  in 
mixing  medicines. 

APOTHECARIES'  FLUID  MEASURE 

60  minims  (n\,)=  1  fluid  dram  (f  3) 

8  fluid  drams  =  1  fluid  ounce  (f  5  ) 
16  fluid  ounces  =  1  pint  (0.) 

8  pints  =  1  gallon  (Cong.) 

One  minim  is  about  equal  to  1  drop 

Exercise  110 

1.  What  part  of  1  gal.  is  1  qt.  ?     1  pt.? 

2.  What  is  the  number  of  cu.  in.  in  1  gal.  ?     In  1  qt.,  liquid 
measure  ?     In  1  pt.,  liquid  measure  ? 

3.  What  part  of  1  bu.  is  1  qt.  ?     1  pt.  ? 

4.  What  is  the  number  of  cu.  in.  in  1  bu.  ?     In  1   qt.,  dry 
measure  ?     In  1  pt.,  dry  measure  ? 

5.  How  many  more  cu.  in.  are  contained  in  1  qt.,  dry  measure, 
than  in  1  qt.,  liquid  measure  ? 


COMPOUND   QUANTITIES  167 

6.  Show  that  1  bu.  is  nearly  equal  to  9.31  gal. 

7.  Find  the  number  of  cu.  ft.  in  1  cd. 

8.  Find  the  number  of  cd.  of  wood  in  a  pile  30  ft.  long,  6  ft. 
high,  and  4  ft.  wide. 

9.  Find  the  cost  of  a  pile  of  wood  24  ft.  long,  5^  ft.  high,  and 
4  ft.  wide,  at  $  6  a  cd. 


UNITS  OF  TIME 
171.    The  prime  unit  of  time  is  1  day. 

MEASURE  OF  TIME 

60  seconds  (sec.)  =  1  minute  (min.) 

60  minutes  =  1  hour  (hr. ) 
24  hours  :  1  day  (da. ) 

7  days  =  1  week  (wk.) 

365  days  =  1  common  year  (yr.) 

366  days  =  1  leap  year  (1.  yr.) 
100  years  =  1  century  (C.) 

The  year  is  divided  into  12  calendar  months : 

January  (Jan.) 31  da.  July 31  da. 

February  (Feb.)     .     .     28  or  29  "  August  (Aug.) 31   " 

March 31  "  September  (Sept.) .     .     .     .  30  " 

April 30  "  October  (Oct.) 31  " 

May 31  "  November  (Nov.)   ....  30  " 

June 30  "  December  (Dec.)    ....  31  " 

In  business  transactions,  1  mo.  is  generally  taken  as  equal  to  30  da.,  and 
1  yr.  as  equal  to  360  da. 

The  following  lines  are  useful  in  enabling  one  to  remember  the  number 

of  days  in  a  month  : 

"  Thirty  days  hath  September, 

April,  June,  and  November." 

A  year  is  the  period  of  the  earth's  revolution  about  the  sun.  It  consists  of 
365  da.  5  hr.  48  min.  50  sec. 

A  common  year  lacks  11  min.  10  sec.  of  being  365  da.  6  hr.,  or  365J  da. 
Hence  when  we  take  365  da.  to  a  common  year,  and  366  da.  to  a  leap 
year,  we  increase  each  year  by  11  min.  10  sec.  In  400  years  this  amounts 
to  a  little  over  3  da.  For  that  reason  three  out  of  four  centennial  years  are 


168  ARITHMETIC 

counted   as  common  years,  i.e.  the   centennial  years  that  do   not   divide 
equally  by  400  have  only  365  da. 

Exercise  111 

1.  Express  9  hr.  as  a  fraction  of  a  wk. 

2.  Express  12  sec.  as  a  decimal  of  a  min. 

3.  Express  146  da.  as  a  fraction  of  a  yr. 

4.  Express  as  a  fraction  of  a  mo. :  10  da. ;  15  da. ;  18  da. 

5.  Express  as  a  decimal  of  a  mo. :  21  da. ;  18  da. ;  27  da. 

6.  Find  the  number-of  da.  between  Jan.  3  and  Feb.  4;  March 
27  and  April  30 ;  Oct.  24  and  Dec.  11. 

7.  How  many  da.  are  there  in  Nov.?    Jan.?   Dec.?    April? 
Feb.? 

8.  State  the  number  of  da.  in  each  of  the  following,  and  also 
what  part  of  a  yr.  each  is,  there  being  30   da.  to  the  mo.  and 
360  da.  to  the  yr. :  1  mo.  10  da. ;  2  mo.  12  da. ;  7  mo.  6  da. 

Exercise  112 

In  the  questions  in  the  following  exercise  add  3  days  to  the 
given  time  (called  days  of  grace)  to  find  the  day  on  which  the 
note  is  due. 

Find  the  date  on  which  a  note  falls  due,  which  I  promise  to  pay  : 

1.  Three  months  after  March  3,  1894. 

2.  Four  months  after  June  13,  1896. 

3.  Ninety  days  after  May  13,  1890. 

4.  Sixty  days  after  Sept.  16,  1895. 

• 

5.  Ninety  days  after  June  4,  1895. 

Find  the  exact  number  of  days  between  the  day  on  which  each 
of  the  following  notes  is  discounted  and  the  day  on  which  it  is 
due: 

6.  Day  of  discount,  May  7,  1894 ;  due  June  6,  1894. 

7.  Day  of  discount,  June  27,  1894;  due  Oct.  16,  1894, 


COMPOUND   QUANTITIES 


169 


8.  Day  of  discount,  Sept.  4,  1894 ;  due  Oct.  30,  1894. 

9.  Day  of  discount,  Dec.  23,  1894 ;  due  Feb.  20,  1894. 
10.    Day  of  discount,  Jan.  15,  1892;  due  May  1,  1892. 

CIRCULAR  OR  ANGULAR  MEASURE 

172.  Angular  Measure  is  used  to  measure  arcs,  angles,  and 
in  determining  latitude,  longitude,  direction,  the  position  of 
vessels  at  sea,  and  the  like. 

173.  A  Circle  is  a  plane  figure  contained  by  one  line  called 
the   circumference,   all   points    of   which  are   equally   distant 
from  a  point  within  it  called  the  centre. 

One-half  of    the    circumference  is   called  the   semicircum- 
ference,  and  one-fourth  a  quadrant. 

An  arc  is  any  portion  of  the  circumference. 

A  line  drawn  through  the  centre  and  terminated  at  both 
extremities  by  the  circumference  is  called  the  diameter. 

The  line  drawn  from  the 
centre  and  terminated  by 
the  circumference  is  called 
the  radius. 

In  the  figure  the  line  OB  has 
revolved  from  OA  through  one- 
fourth  of  a  revolution.  The  an- 
gle AOB  is  called  a  right  angle 
and  contains  90°. 

OA  and  OB  are  said  to  be  per- 
pendicular to  each  other. 

ANGULAR  MEASURE 

60  seconds  (")  =  1  minute  (') 
60  minutes        =  1  degree  (°) 
360  degrees         =  1  circumference  (C.) 

The  circumference  of  the  earth  at  the  equator  =  24,902  mi. 

The  length  of  a  degree  at  the  equator  =  24,902  mi.  -=-  360  =  69.17  mi. 


170  ARITHMETIC 

Exercise  113 

1.  What  part   of    a  revolution    is    1    right  angle  ?     2    right 
angles  ?     4  right  angles  ? 

2.  What  part  of  a  revolution  is  60°  ?     45°  ?     225°  ? 

3.  What  is  the  length  of  the  arc  of  1°  in  a  circle  whose  cir- 
cumference is  360  yd.  ? 

4.  If  an  arc  of  3°  is  6  ft.  long,  what  is  the  length  of  the  cir- 
cumference of  the  circle  ? 


MISCELLANEOUS  UNITS 


174.  NUMBERS 


12  units  =  1  dozen  (doz.) 
12  dozen  =  1  gross 
12  gross  =  1  great  gross 
20  units  =  1  score 


PAPER 


24  sheets  =  1  quire 
20  quires  —  1  ream 
2  reams     =  1  bundle 
5  bundles  =  1  bale 


MISCELLANEOUS  WEIGHTS 

175.  A  bushel  of  wheat  =  60  Ib. 

A  bushel  of  beans  =  60  Ib. 

A  bushel  of  clover  seed  =  60  Ib. 
A  bushel  of  shelled  corn  =  56  Ib. 
A  bushel  of  rye  =  56  Ib. 

A  bushel  of  barley  =  48  Ib. 

A  bushel  of  oats  =  32  Ib. 

A  bushel  of  potatoes  —  60  Ib. 
A  bushel  of  coarse  salt  (domestic)  =  56  Ib. 

These  are  the  legal  number  of  pounds  per  bushel  in  Michigan,  Indiana, 
Illinois,  Wisconsin,  Iowa,  Missouri,  and  New  York. 

On  the  Chicago  Board  of  Trade  seeds  are  sold  by  the  cental, 

A  barrel  of  flour  =  196  Ib. 

A  barrel  of  pork  or  beef  =  200  Ib. 
A  cental  of  grain  =  100  Ib. 

Exercise  114 

1.    Pens  are  sold  in  boxes  containing  1  gross.     How  many  pens 
are  there  in  a  box  ? 


COMPOUND   QUANTITIES  171 

2.  Lead  pencils  are  sold  in  boxes  containing  i  gross.     How 
many  pencils  are  there  in  a  box  ? 

3.  How  many  packages  of   lead  pencils  of   1   doz.  each  are 
there  in  a  box  ? 

4.  Eggs  are  packed  in  crates  holding  30  doz.     How  many  eggs 
in  a  crate  ? 

5.  How  many  sheets  of  paper  in  20  quires  ? 

6.  What  articles  of  food  weigh  60  Ib.  to  the  bu.  ? 

7.  How  many  bu.  of  wheat  weigh  as  much  as  10  bu,  of  bar- 
ley  ?  • 

8.  What  is  the  ratio  of  the  weight  of  1  bu.  of  barley  to  that 
of  1  bu.  of  oats  ? 

176.  A  certain  room  is  8  yd.  long.  Here  the  unit  of 
measurement  is  1  yd.,  and  the  measure  of  the  length  of 
the  room  is  the  number  8. 

The  area  of  the  floor  of  a  room  is  192  sq.  yd.  6  sq.  ft. 
Here  the  measure  of  the  area  is  the  sum  of  192  units  of 
1  sq.  yd.  and  6  units  of  1  sq.  ft. 

A  pitcher  holds  |  of  a  gal.  of  water. 

Here  the  measure  of  the  capacity  of  the  pitcher  is  |  and 
the  unit  of  measurement  is  1  gal. 

Exercise  115 

Name  the  measures  of  the  following  quantities,  and  the  units 
of  measurement : 

1.  The  volume  of  a  cistern  which  holds  450  cu.  ft. 

2.  The  volume  of  a  cistern  which  holds  1200  gal. 

3.  The  area  of  a  field  which  contains  8J  A. 

4.  The  value  of  a  house  worth  $  2800.     What  are  the  meas- 
ures of  the  value  of  the  house  with  the  following  units  :  $  5,  $  10, 
$  50,  $  100  ? 


172  ARITHMETIC 

5.  The  weight  of  200  Ib.  of  sugar.    What  are  the  measures  of 
the  weight  with  1  oz.,  1  cwt.,  and  1  T.  as  units  ? 

6.  The  weight  of  a  quantity  of  tea  weighing  336  Ib.     If  1 
long  cwt.  is  used  as  the  unit,  what  is  the  measure  ? 

7.  The  weight  of  8  oz.  of  gold.     If  1  pwt.  is  the  unit  of  meas- 
urement, what   number  expresses  the  measure  ?     What  number 
measures  the  weight  when  1  Ib.  is  the  unit  ? 

8.  The  weight  of  40  gr.  of  quinine.     What  is  the  measure 
when  1  ±)  is  the  unit? 

9.  What  are  the  measures  of   1  Ib.  of   gold,  of  lead,  and  of 
quinine,  when  the  unit  of  measurement  is  1  oz.?      When  1  gr. 
is  the  unit  ? 

10.  The  length  of  the  circumference  of  a  circle  found  to  be  22 
yd.  long.     What  are  the  measures  of  the  circumference  when  1  ft. 
and  1  rd.  are  the  measures? 

11.  The   length   between   two   points  which   measures   4   ch. 
What  are  the  measures  of  the  length  when  the  units  are  1  mi. 
and  1  link  ? 

12.  The  area  of  a  field  which  contains  80  sq.  rd.     What  are 
the  measures  of  the  field  when  the  units  are  1  A.  and  1  sq.  yd.  ? 

13.  The  capacity  of  a  pitcher  which  contains  J  of  a  gal.  of 
water.     What  is  the  measure  when  the  unit  is  1  qt.  ? 

14.  The  capacity  of  a  basket  which  holds  6|  qt.     What  are 
the  measures,  the  units  being  1  pt.  and  1  pk.  ? 

15.  The  time  of  a  rainstorm,  which  lasted  2  hr.     What  are 
the  measures,  the  units  being  1  min.  and  1  da.  ? 

16.  The  weight  of  a  silver  cup  which  is  15  oz.  12  pwt.  12  gr. 

177.    The  fundamental  units  used  in  the  measurement  of 
value  are  1  cent,  1  dime,  1  dollar,  and  1  eagle. 


COMPOUND   QUANTITIES  173 

The  value  of  a  postage-stamp  used  to  mail  a  letter  to  any 
part  of  the  United  States  or  Canada  is  measured  by  the 
number  2  and  the  unit  1  ct. 

The  cost  of  a  quart  of  berries  worth  15  ct.  is  measured  by 
the  number  3  and  the  unit  1  nickel,  or  by  the  number  1 
and  the  unit  1  dime  plus  the  number  1  and  the  unit  the 
nickel. 

It  may  also  be  measured  by  the  number  15  and  the  unit  1 
ct.  It  may  also  be  measured  by  the  number  1  and  the  unit 
1  quarter-dollar  less  the  number  1  and  the  unit  1  dime. 

The  unit  for  measuring  oil  is  1  gal.  That  for  buying  spice 
by  retail  is  1  oz.  Frequently  a  quantity  is  expressed  with 
reference  to  two  or  more  units.  Thus,  the  length  of  a  table 
being  1  yd.  2  ft.  6  in.,  the  units  are  1  yd.,  1  ft.,  and  1  in. 


Exercise  116 

1.  Name  instances  in  which  $  1  is  the  unit  of  value;  1  ct. ; 
1   nickel;    1    dime.      What    unit    of    value    is   most   commonly 
used  ? 

2.  Name  instances  in  which  the  units  of  weight  used  are  1 
oz. ;  1  Ib. ;  l.cwt. ;  1  T. 

3.  Name  quantities  whose  weight  is  measured  by  these  units  : 
1  gi\,  1  pwt.,  1  oz.,  and  1  Ib. 

4.  Name  quantities  which  are  measured  by  the  units :  1  gr., 
13,  13,  IS,  1  Ib. 

5.  What  quantities  are  expressed  in  terms  of  these  units :  1 
in.  ?  1  ft.  ?  1  yd.  ?  1  rd.  ?  1  mi.  ? 

6.  Name  things  whose  measurement  is  given  in  terms  of  the 
units  1  pt.,  1  qt.,  1  gal. 


174  ARITHMETIC 

7.  Name  articles  whose  measurement  is  expressed  in  terms  of 
the  unit  1  bu. ;  1  pk. ;  1  qt. 

8.  In  measuring  time  give  instances  in  which  you  use  as  the 
unit  1  century ;  1  yr. ;  1  mo. ;  1  wk. ;  1  da. ;  1  hr. ;  1  min. ;  1  sec. 

9.  Name  articles  whose  quantity  is  expressed  in  terms  of  the 
unit  1  doz. ;  1  gross ;  1  great  gross ;  1  score. 

10.  In  measuring  paper  the  following  units  are  used :  1  quire, 
1  ream.     Give  instances  when  1  quire  is  used  as  the  unit,  and 
also  when  1  ream  is  used. 

11.  What  unit   of   weight   connects   Avoirdupois,   Troy,   and 
Apothecaries'   weight  ?      What   number   expresses   the   measure 
of   1   Ib.    of    each   kind   in   terms   of    the   common   unit?      Of 
1  oz.  ? 

12.  What  units  are  common  to  Apothecaries'  and  Troy  weight 
and  of  equal  value  ? 

13.  What  units  of   length  connect   Surveyors'  Long  Measure 
with  Linear  Measure  ? 

14.  Name  five  units  of  area  which  are   derived  from  corre- 
sponding units   of  length.      Why   is   1  A.  chosen   as  a  unit  of 
area?      Give  instances   in.  which  1   A.   is  used  as  the  unit  of 
area,  and  also  when  1  sq.  mi.  is  the  unit. 

15.  Name  a  unit  of  volume  larger  than  1  cu.  yd.      Why  have 
we  no  units  of  volume  corresponding  to  the  linear  units,  1  rd. 
and  1  mi.  ? 

16.  Find   the   number   of   cu.  in.  in  1  qt.,  dry  measure,  and 
find  how  much  greater  it  is  than  1  qt.,  liquid  measure. 

17.  State  the  number  of  days  in  the  years  1500,  1600,  1700, 
1800,  1900,  2000. 

18.  The  circumference  of  a  circle  is  1,296,000  in.  in  length. 
Find  the  length  of  1°;  of  1';  of  1". 


COMPOUND   QUANTITIES  175 

REDUCTION 

178.  Reduction  Descending  is  the  process  of  reducing  a 
quantity  expressed  in  terms  of  a  unit  or  units  of  measure- 
ment to  a  quantity  expressed  in  terms  of  a  smaller  unit,  or  of 
smaller  units  of  measurement. 

Reduce      2  mi.  36  rd.  5  yd.  2  ft.  to  feet. 
320 

640 
36 

676  rd. 

NOTE.  —  In  this  reduction  we  are  to  think 
of  the  operation  as  signifying  2  x  320  rd.,  or 
5  by  the  law  of  commutation  as  320  x  2  rd.,  and 


3723  yd.        not  as  representing  320  x  2  mi. 
3 


11169 
2 

11171  ft. 

2  mi.  =  2  x  320  rd.  =  640  rd. 
2  mi.  36  rd.  =  640  rd.  +  36  rd.  =  676  rd. 

676  rd.  =  676  x  5£  yd.  =  3718  yd. 
2  mi.  36  rd.  5  yd.  =  3718  yd.  +  5  yd.  =  3723  yd. 

3723  yd.  =  3723  x  3  ft.  =  11,169  ft. 
/.  2  mi.  36  rd.  5  yd.  2  ft.  =  11,169  ft.  +  2  ft.  =  11,171  ft. 

Exercise  117 

1.  Eeduce  5  gal.  3  qt.  1  pt.  2  gi.  to  gills. 

2.  Keduce  18  bu.  6  pk.  3  qt.  1  pt.  to  pints. 

3.  Keduce  £5  12s.  9cZ.  to  pence. 

4.  Reduce  16  Ib.  8  oz.  15  pwt.  17  gr.  to  grains. 

5.  Reduce  7  T.  18  cwt.  14  Ib.  12  oz.  to  ounces. 

6.  Keduce  2  yr.  15  da.  17  min.  to  minutes. 

7.  Keduce  18  rd.  4  yd.  2  ft.  6  in.  to  inches. 

8.  Keduce  12  Ib.  5  oz.  5  dr.  2  sc.  16  gr.  to  grains. 

9.  Keduce  2  A.  4  sq.  rd.  2  sq.  yd.  8  sq.  ft.  to  square  feet. 


176  ARITHMETIC 

10.  Reduce  16  cu.  ft.  1374  cu.  in.  to  cubic  inches. 

11.  Reduce  1  mi.  18  rd.  2  yd.  2  ft.  6  in.  to  inches. 

12.  State  how  to  reduce  a  quantity  from  higher  to  lower  de- 
nominations. 

13.  Find  the  number  of  A.  in  a  township. 

14.  Find  the  number  of  cu.  in.  in  a  vessel  containing  25  gal. 

15.  Reduce  59  Ib.  7  oz.  14  pwt.  19  gr.  to  grains. 

16.  Reduce  7  T.  15  cwt.  56  Ib.  to  ounces. 

17.  Reduce  17  ft.  2  5  23  to  grains. 

18.  Reduce  17  cu.  yd.  1001  cu.  in.  to  cubic  inches. 

19.  Reduce  760  bu.  3  pk.  to  quarts. 

20.  Reduce  56  reams  19  quires  18  sheets  to  sheets. 

21.  Reduce  36  wk.  5  da.  17  hr.  to  seconds ;  and  1  mo.  of  30  da. 
23  hr.  59  sec.  to  seconds. 

22.  Reduce  35  A.  80  sq.  rd.  28  sq.  yd.  to  square  yards. 

179.  Reduction  Ascending  is  the  process  of  reducing  a 
quantity  expressed  in  terms  of  a  unit,  or  of  units,  to  a  quan- 
tity expressed  in  terms  of  a  larger  unit,  or  of  larger  units. 

(1)    Reduce  242,337  in.  to  higher  denominations. 


12 


11 

320 


242337 


20194  ft.  9  in. 


6731  yd.  1  ft. 
2 


13462  half  yd. 


1223  rd.  9  half  yd.,  i.e.  4  yd.  1  ft.  6  in. 


3  mi.  263  rd. 

.-.  242,337  in.  :=  3  mi.  263  rd.  4  yd.  1  ft.  9  in. ) 

1  ft.  6  in.  / 
=  3  mi.  263  rd«.  5  yd.  0  ft.  3  in. 


COMPOUND   QUANTITIES  177 

242,337  in.  =  20,194  ft.  9  in. 
20,194  ft.  9  in.  =  6731  yd.  1  ft.  9  in. 
6731  yd.  1  ft.  9  in.  =  1223  rd.  4J  yd.  1  ft.  9  in. 
1223  rd.  4£  yd.  1  ft.  9  in.  =  3  mi.  263  rd.  4J  yd.  1  ft.  9  in.' 

=  3  mi.  263  rd.  5  yd.  0  ft.  3  in. 
.-.  242,337  in.  =  3  mi.  263  rd.  5  yd.  3  in. 

To  prove  this  answer  correct,  reduce  3  mi.  263  rd.  5  yd.  3  in.  to 
in.,  by  the  method  of  the  preceding  exercise. 

ISO.  We  reduce  a  quantity  expressed  in  terms  of  a  smaller  unit  to  larger 
units  in  order  to  get  a  more  definite  idea  of  its  value. 

Thus  we  form  no  definite  idea  of  a  distance  between  two  points  when  we 
are  told  that  it  is  242,337  in.  ;  but  we  have  a  definite  conception  of  the 
same  distance  when  we  are  told  that  it  is  3  mi.  263  rd.  5  yd.  3  in. 

Exercise  118 

Reduce  to  higher  denominations,  and  prove  every  third  answer 
correct : 

1.  678  pt.  4.    4728  cu.  ft.  to  cd. 

2.  4622  pt.  of  dry  measure.          5.    18,425  Ib.  of  wheat  to  bu. 

3.  483,197  sec.        .  6.    21,489  d.     . 

7.  93,742  oz. 

8.  State  how  to  reduce  a  quantity  expressed  in  terms  of  a 
lower  unit  to  higher  units. 

9.  Reduce  5420  gr.  to  oz.,  Avoir. 

10.  141,728  gr.  14.  1,364,428  in. 

11.  57,893  cu.  in.  15.  273,460  sq.  yd. 

12.  56,735  d  16.  6,188,724  sq.  in. 

13.  5,838,297  oz.  17.  429,678  in. 

18.  73,940  TV 

19.  89,673  gr.,  Apothecaries'  weight. 

20.  7493  units. 

21.  Reduce  37,921  in.  to  ch.,  rd.,  etc. 

N 


178  ARITHMETIC 

22.  Eeduce  121,838  A.  to  townships,  etc. 

23.  The  Imperial  gal.  of  Great  Britain  contains  277.274  cu.  in. 
Find,  correct  to  three  decimal  places,  the  number  which  measures 
the  Imperial  gal.,  the  gal.,  liquid  measure,  being  the  unit. 

24.  Express  in  bu.  and  cu.  in.  the  volume  of  a  bin  8  ft.  by  5  ft. 
by  4  ft. 

25.  How  many  cu.  in.  are  there  in  4  qt.,  dry  measure  ?     Show 
that  this  is  .163  greater  than  the  gal.  measure. 

COMPOUND  ADDITION  AND  SUBTRACTION 
181.   Add: 


mi. 

rd. 

yd. 

ft. 

in. 

2 

27 

1 

2 

8 

1 

146 

2 

1 

6 

8 

91 

2 

0 

4 

7 

152 

1 

2 

9 

19 

97 

2i 

1 

3 

i 

2 

=    1 

6 

19  97  2  2  9 

The  sum  of  the  in.  column  is  27  in.,  or  2  ft.  3  in. 

The  sum  of  the  ft.    column,  increased  by  2  ft.,  is  7  ft.,  or  2  yd:  1  ft. 

The  sum  of  the  yd.  column,  increased  by  2  yd.,  is  8  yd.,  or  1  rd.  2i  yd. 

The  sum  of  the  rd.  column,  increased  by  1  rd.,  is  417  rd.,  or  1  mi.  97  rd. 

The  sum  of  the  mi.  column,  increased  by  1  mi.,  is  19  mi. 

Changing  L  yd.  to  1  ft.  6  in.  and  adding,  we  have  the  sum  =  19  mi.  97  rd. 
2  yd.  2  ft.  9  in. 

As  in  the  problems  in  addition  in  Chapter  III.,  we  are  required  in  the 
preceding  question  to  find  the  whole  quantity  measured  by  the  four  given 
parts,  of  which  the  first  is  2  mi.  27  rd.  1  yd.  2  ft.  8  in.  What  are  the  other 
three  measured  parts  ? 

In  this  question  how  would  the  work  of  writing  and  adding  be  diminished 
if  our  units  of  length  were  arranged  according  to  the  decimal  system  ? 


COMPOUND   QUANTITIES  179 

182.   Subtract  53  Ib.  5  oz.  18  pwt.  from  72  Ib.  4  oz.  7  pwt. 

lb.  oz.  pwt. 

72  4  7 

53  5  18 


18          10  9 

Since  we  cannot  take  18  pwt.  from  7  pwt.,  take  1  oz.  or  20  pwt.  from  4  oz. 
and  add  it  to  the  7  pwt.,  making  27  pwt.  18  pwt.  from  27  pwt.  leaves  9  pwt. 
Since  we  cannot  take  5  oz. ,  from  3  oz.,  take  1  lb.  or  12  oz.  from  72  lb.  and 
add  it  to  the  3  oz.,  making  15  oz.  5  oz.  from  15  oz.  leaves  10  oz.  53  lb. 
from  71  lb.  leaves  18  lb. 

Hence  the  difference  =  18  lb.  10  oz.  9  pwt. 

As  in  the  problems  in  subtraction  in  Chapter  IV.,  we  are  given  in  the 
preceding  question  the  whole  quantity  measured  by  72  lb.  4  oz.  7  pwt.,  and 
one  part,  viz.  53  lb.  5  oz.  18  pwt.,  and  are  required  to  find  the  part  measured 
by  their  difference. 

Exercise  119 


Add: 

bu. 

Pk. 

qt. 

pt. 

1.    3 

5 

6 

1 

8 

4 

1 

0 

7 

3 

5 

1 

9 

4 

3 

1 

lb                   oz.  pwt 

2.    18             11  16 

16              9  22 

23              8  6 
6 


4. 

T. 

16 

cwt. 

17 

lb. 

74 

13 

10 

20 

17 

15 

19 

84 

0 

87 

5. 

11 

11 

36 

3 

22 

3 
3 

9 

2 

gr. 
19 

56 

0 

1 

10 

3 

2 

2 

11 

15 

6 

1 

9 

79 

4 

1 

10 

£  *.  d. 

3.     5  17  10  cu.  yd.  cu.  ft.  cu.  in. 

36  0  11 

734 

73  19  8 

30  14  5 


6.  3 

23 

171 

17 

17 

31 

28 

26 

1000 

34 

23 

1101 

180  ARITHMETIC 

7.    State  how  to  add  compound  quantities. 
Subtract : 


lb. 

oz. 

pwt. 

ft)    5    3   9 

8. 

144 

8 

14 

10.  144    9    4    1 

106 

11 

16 

129    0    7    3 

yd. 

ft. 

in. 

lb.      oz.     pwt.     gr. 

9. 

15 

1 

5 

11.  5836    000 

13 

2 

7 

4976    7    13    19 

cu.  yd. 

cu.  ft.       cu.  in. 

12. 

37 

18      857 

35 

24     1280 

13.  State  how  to  subtract  one  compound  quantity  from  another. 

14.  State  in  what  respects  addition  and  subtraction  of  com- 
pound quantities  are  the  same  as  addition  and   subtraction  of 
numbers,  and  state  how  they  differ. 

COMPOUND  MULTIPLICATION  AND  DIVISION 

183.   Multiply  5  wk.  6  da.  18  hr.  by  11. 

We  are  here  required  to  find  the  whole  quantity  measured  by  the  unit 
5  wk.  6  da.  18  hr.  and  the  number  11. 

wk.      da.      hr. 

5      6      18 
11 


65      4        6 

We  multiply  18  hr.  by  11  and  obtain  the  product  198  hr.  or  8  da.  6  hr. 
Then  we  multiply  6  da.  by  11  and  obtain  the  product  66  da.,  which, 
increased  by  8  da.  is  74  da.,  or  10  wk.  4  da.  Multiplying  5  wk.  by 
11,  and  adding  10  wk.,  we  have  65  wk.  Hence  the  product  is  65  wk. 
6  da.  18  hr. 

What  is  the  ratio  of  65  wk.  4  da.  6  hr.  to  5  wk.  6  da.  18  hr.  ? 


COMPOUND   QUANTITIES  181 

184.    (1)  Divide  88  rd.  3  yd.  1  ft.  by  34. 

rd.    yd.    ft.    rd.  yd.   ft. 

34)  88     3     1   (2     3     1 
68 

20  rd. 

_5_i 

110  yd. 
_3 

113  yd. 
102 

llyd. 

_3 

33ft. 
_1 

34ft. 
34 

Hence  the  quotient  or  unit  of  measure  is  2  rd.  3  yd.  1  ft. 

The  remainder  on  dividing  88  rd.  by  34  is  20  rd.  20  rd.  3  yd.  =113  yd. 
The  remainder  on  dividing  113  yd.  by  34  is  11  yd.  11  yd.  1  ft.  =  34  ft. 

On  dividing  34  ft.  by  34,  there  is  no  remainder.  Hence  the  quotient  is 
2  rd.  3  yd.  1  ft.  What  part  of  the  dividend  is  the  quotient  ? 

(2)  Divide  73  gal.  1  pt.  by  16  gal.  1  qt. 

We  are  here  required  to  find  the  number  which  is  the  ratio  of  the  quan- 
tity 73  gal.  1  pt.  to  the  unit  16  gal.  1  qt. 

73  gal.  1  pt.  =  585  pt.  ;  16  gal.  1  qt.  =  130  pt. 

585  pt.  -r-  130  pt.  =  4i. 
.•.  the  quotient  or  ratio  is  4£. 

Exercise  120 

1.  Multiply  7  gal.  3  qt.  1  pt.  by  9. 

2.  Multiply  8  da.  12  hr.  25  min.  by  28. 

3.  Divide  £199  6s.  8  d.  by  13. 

4.  Divide  459  Ib.  4  oz.  5  pwt.  22  gr.  by  29. 

5.  Multiply  86  Ib.  7  oz.  16  pwt.  11  gr.  by  8. 

6.  Multiply  5  wk.  6  da.  18  hr.  14  min.  by  11, 


182!  AKJ'JHMJ/nC 

7.  Divide  JT.'iS  cu.  yd,   JL'W  CU,  in.  by  708. 

8.  Divide  081  da.  8  hr.  'J  min.  by  17. 

9.  .Mu  Hi  ply  70  yd.  2  R.  10  in.  by  7. 

10.  Divide  1  mi,  ~A  rd.  1  ft.  2  in.  by  29. 

11.  Multiply  2  hr.  8  nun.  0  son.  by  15. 
.    154.    Divide  13°  26'  by  15. 

13.  Divide  11  yd.  1   R.  8  in.  by  .'J  yd.  \  in. 

14.  Divide,  10  |)ii.  1   pk.  2  fjl,.  by  10  bu.  .",  pk.  4  qt. 

15.  (a;   State  how  to  multiply  ;i,  compound  rpianl-Hy  by  a  given 

nnji]b':r.    ('/yj  St;ii<-  how  lo  divide  a  compound  quantity  l>y  a  given 
Dumber, 

\(\.     Sl;it«-  in  what  i'<:spf:cts  multiplication  :md  division  ol'  com- 
pound fjii;uititi(;s  ;u'<;  like  niultij>lic;i.tion  and  division  of  nmnbc 
and  how  they  differ. 

FRACTIONS  OF  SIMI-LK  AND  COMJ-OHXO  QUANTITIES 
185.    C  \  )    Find  tljc  value  of  •'.  of  a  mi. 

i;  rni.  -  \  of  320  M.  =  20«:^  nl. 
ij  r'l.  ;;  r,f  .0^  yd.  yj  yd. 
J  yd.  :  [of  o  ft,  2  ft. 

.-.  ,';  of  ami.  =±  200  nl.  .'i  yd.  2  ft. 

C2)   Find  the  value  of  f;  1m.  -  -  f>  pk. 

g  bu.  -     |of  4pk.  :     2';';  pk. 

2fpk.  -    :•  pk.  -    if|  Pk. 

?Jpk.=    -:  of  Hqt.  -4,.';  qt, 

.'.      fj    bU.    --    §    ]jk.    r:    ]    pk.    4,/1,    qt. 

Exercise  121 

Irind  the  value  of  : 


1.  J  of   JJl  ;    2  of  ,*;2. 

2.  ,:;0  of  a  da.  ;    ,r',  of  ;i,  mi. 

3.  J  of  :j  T.  ;  7|  Ib.  Avoir. 


COMPOUND   QUANTITIES  183 

4.  |  bu.  -  -  f  pk. ;  f  Ib.  Troy  +  f  Ib.  Troy    -  f  oz.  Troy. 

5.  f  A.  5  I  A.  -  -  -|  sq.  rd. ;  f  sq.  mi. 

6.  f  of  5  Ib.  8  oz.  6  pwt. 

7.  f  of  2  mi.  38  rd.  4  yd.  2  ft.  2  in. 

8.  State  how  to  express  a  fraction  of  a  unit  of  measure  in 
trims  of  smaller  units.       • 

Express  .854  of  an  A.  in  lower  denominations. 

.854  A. 
160 


51240 
854 

136.640  sq.  rd. 
30 1 

16 
1920         .  •.  .854  A.  -  136  sq.  rd.  19  sq.  yd.  3  sq.  ft.  34.56  sq.  in. 

19.36  sq.  yd.   Adapt  note,  §  178,  to  this  problem. 
9 

3.24  sq.  ft. 
144 

96 
96 
24 

34.56  sq.  in. 

Exercise  122 

Find  the  value  of : 

1.  .84  of  a  da.  -  4.    5.923  mi.  -  75.18  rd. 

2.  .045  of  a  mi.  5.    £  75.43  -  16.76s. 

3.  .6  of  a  Ib.  Troy.  6.    4.7  A.  --2.93  sq.  rd. 

186.    Reduce  213  rd.  5  ft.  6  in.  to  the  fraction  of  3  ini. 

213  rd.  5  ft.  6  in.  =  42,240  in. 

3  ini.  -=  3  x  63,360  in.  =:  190,080. 

.-.  213  rd.  5  ft.  6  in.  =  iWoVu,  or  I  of  3  mi- 

The  G.  C.  M.  of  42,240  and  190,080  is  21,120,  which  divides  the  numera- 
tor twice  and  the  denominator  9  times. 


184  ARITHMETIC 

Exercise  123 

1.  Reduce  £  1  7s.  6<t  to  the  fraction  of  £  2. 

2.  Reduce  22  rd.  5  yd.  2  ft.  6  in.  .to  the  fraction  of  a  mi. 

3.  Reduce  8  hr.  3  min.  to  the  fraction  of  a  da. 

4.  Reduce  4  mo.  3  da.  to  the  fraction  of  a  yr.  (30  da.  to  a 
mo.  and  360  da.  to  a  yr.). 

5.  Reduce  11  mo.  18  da.  to  the  fraction  of  a  yr. 

6.  Reduce  1J  in.  to  the  fraction  of  11  yd. 

7.  Reduce  J  Ib.  Avoir,  to  the  fraction  of  2  Ib.  Troy. 

8.  Express  213  rd.  1  yd.  2  ft.  6  in.  as  a  fraction  of  145  rd. 
2  yd.  1  ft.  6  in. 

187.   Express  58  rd.  2  yd.  7.2  in.  as  a  decimal  of  a  mi. 


12 
3 

5.5 
320 

7.2  in. 

.6  ft. 

2.2  yd. 

58.4  rd. 

.  1825  mi. 

7.2  in.  =  .6  ft.  =  .2  yd. 
2  yd.  +  .2  yd.  =  2.2  yd.  =  .4  rd. 

58.4  rd.  =  .1825  mi. 
.-.  58  rd.  2  yd.  7.2  in.  =  .1825  mi. 
Prove  this  answer  correct  by  reducing  .1825  mi.  to  lower  denominations. 


Exercise  124 

1.  Reduce  8  oz.  15.2  pwt.  to  the  decimal  of  a  Ib. 

2.  Reduce  21  hr.  57  min.  36  sec.  to  the  decimal  of  a  da. 

3.  Reduce  147  rd.  1  yd.  3.6  in.  to  the  decimal  of  a  mi. 

4.  25  sq.  mi.  128  A.  is  what  decimal  of  a  township  ? 

5.  Reduce  67  sq.  rd.  6  sq.  yd.  64.8  sq.  in.  to  the  decimal  of 


an  A. 


COMPOUND   QUANTITIES  185 

BOARD  MEASURE 

188.  Boards  1  inch  or  less  in  thickness  are  sold  by  the 
square  foot. 

Thus  a  board  18  ft.  long,  14  in.  wide,  and  1  in.  thick  or  less  contains 
18  x  ^f,  or  21  ft.,  board  measure. 

To  find  the  number  of  feet,  board  measure,  in  lumber  more 
than  1  inch  thick,  we  find  the  number  of  square  feet  in  the 
surface  of  the  board  and  multiply  this  result  by  the  number 
of  inches  that  the  lumber  is  thick. 

Thus  a  board  15  ft.  long,  8  in.  wide,  and  2£  in.  thick  contains  15xT82X  |, 
or  25  board  feet. 

Exercise  125 

How  many  feet,  board  measure,  in : 

1.  A  board  20  ft.  long,  9  in.  wide,  and  1  in.  thick  ?   f  in.  thick  ? 

2.  A  board  18  ft.  long,  8  in.  wide,  and  2J  in.  thick  ? 

3.  A  scantling  16  ft.  long,  3  in.  wide,  and  4  in.  thick  ? 

4.  Twenty  scantlings,  24  ft.  long,  5  in.  wide,  and  7  in.  thick  ? 

5.  A  stick  of  timber  33  ft.  long,  and  14  in.  square  ? 

6.  One  cu.  ft.  ? 

7.  What  is  the  cost  of  25  joists  each  6  in.  by  4  in.  by  15  ft. 
at  $ 22  per  M.  ? 

8.  What  is  the  cost  of  24  joists  each  5  in.  by  7  in.  by  10  ft. 
at  $  21  per  M.  ? 

9.  How  much  will  it  cost  to  enclose  a  rectangular  lot  50  ft. 
wide  and  100  ft.  deep  with  a  tight  board  fence  16  ft.  high  with 
boards  that  cost  $  18  per  M.  ? 

LONGITUDE  AND  TIME 

189.  Turn  to  your  Geography  and  find  several  meridian 
lines.    Find  the  prime  meridian  which  passes  through  Green- 
wich, England. 


186  ARITHMETIC 

The  imaginary  lines  drawn  on  the  earth's  surface  from 
pole  to  pole  are  called  meridians.  The  meridian  passing 
through  Greenwich,  a  town  near  London,  England,  hav- 
ing the  royal  observatory,  is  called  the  prime  or  standard 
meridian. 

Places  west  of  the  prime  meridian  are  in  west  longitude, 
and  places  east  of  the  prime  meridian  are  in  east  longitude. 
Thus,  Washington  is  77°  T  west  longitude,  and  Paris  2°  20' 
east  longitude. 

190.  Find    from   the    maps   in   your   geographies   to   the 
nearest   degree    the   longitude   of   these  cities :    New  York, 
Pittsburg,  Richmond,  Atlanta,  Chicago,  Denver,  Salt  Lake 
City,  San  Francisco. 

Find  also  the  longitude  of  Rome,  Stockholm,  Athens,  Con- 
stantinople, St.  Petersburg,  and  Moscow. 

191.  The    difference   in   longitude  between   Philadelphia, 
which  is  75°  9'  west  longitude,  and  Portland,  which  is  70°  15' 
west  longitude,  is  4°  54'. 

The  difference  between  the  longitude  of  Philadelphia  and 
that  of  Paris,  which  is  2°  20'  east  longitude,  is  77°  29',  and  is 
obtained  by  finding  the  sum  of  the  longitudes. 

192.  Find  on  the  map  of  the  United  States  and  name  the 
meridians  that  denote  Eastern  time,  Central  time,  Mountain 
time,  and  Pacific  time.    How  many  degrees  are  there  between 
these  meridian  lines?   What  is  the  difference  in  time  between 
places  situated  on  these  meridians? 

193.  As  the  sun  rises  in  the  east,  it  is  sunrise  in  New  York 
earlier  than  in  Chicago ;   consequently  at  any  time  during 
the  day  the  clock  time  in  New  York  is  later  than  in  Chicago. 


COMPOUND   QUANTITIES  187 

Similarly,  clock  time  in  San  Francisco,  which  is  west  of 
Chicago,  is  earlier  than  in  the  latter  city. 

194.  Since  the  sun  appears  to  move  in  a  circle  about  the 
earth,  i.e.  through  360°  in  24  hr.,  we  have  the  following: 

In  24  hr.  the  sun  passes  through  360°. 

In    1  hr.  the  sun  passes  through  15°. 

In  1  min.  the  sun  passes  through  -fa  of  15°  =  i°  =  15'. 

In  1  sec.  the  sun  passes  through  -fa  of  15'  =  \'  =  15". 

Hence,  to  reduce  longitude  expressed  in  time  to  longitude 
expressed  in  degrees,  ive  multiply  by  15.  and  to  reduce  longi- 
tude expressed  in  degrees  to  longitude  expressed  in  time,  ive 
divide  by  15. 

195.  Make  the  multiplication  table  of  15  and  memorize  it,  so  as  to  be 
able  to  work  questions  in  longitude  and  time  by  short  multiplication  and 
division. 

Exercise  126 

1.  What   is   the  difference  in  longitude  between  two  places 
whose  difference  in  time  is   1   hr.  ?     2  hr.  ?     4  hr.  ?     2  rnin.  ? 
3  min.  ?     1  sec.  ?     3  sec.  ? 

2.  What   is  the  difference  in  longitude  between   two  places 
whose  difference  in  time  is  1  hr.  2  min.  ?     1  hr.  3  min.  2  sec.  ? 

3.  What  is  the  difference  in  time  between  two  places  whose 
difference  in  longitude  is  30°  ?    75°?    120'?    90'?    135"?    105"? 

4.  What  is  the  difference  in  time  between  two  places  whose 
difference  in  longitude  is  15°  45'  30"  ?     75°  15'  45"  ? 

5.  Find   the   difference   in   longitude   between   the   following 
places,  and  illustrate  your  answers  by  diagrams : 

Washington  77°  west  longitude  and  Helena  112°  west  longitude  ? 
Washington  77°  west  longitude  and  Hamburg  10°  east  longitude  ? 
Cairo  32°  east  longitude  and  Hamburg  10°  east  longitude  ? 


188  ARITHMETIC 

6.  What  is  the  difference  in  time  between  two  places : 

(1)  One  64°  west  longitude,  the  other  34°  east  longitude  ? 

(2)  One  64°  west  longitude,  the  other  26°  east  longitude  ? 

(3)  One  64°  east  longitude,  the  other  34°  east  longitude  ? 

7.  When   it  is  6   A.M.  at   San  Francisco,   what  time  is  it  at 
a  place   45°   east  of   San  Francisco?     30°  east?     15° 45'  east? 

30°  15' 45"  east? 

8.  When  it  is  11  A.M.  at  Chicago,  what  time  is  it  at  a  place 
60°  west  of  Chicago  ?     30°  west  ?     45'  west  ?     15°  45'  ? 

196.  (1)  Find  the  difference  in  time  between  St.  Louis  90°  19' 
26"  west  longitude  and  Sacramento  121°  25' 41"  west  longitude. 

121°    25'  41" 

90      19  26 

15)31°       6'  15"        difference  in  longitude 

2  hr.  4  min  25  sec.  difference  in  time 

A  difference  in  longitude  of  31°  gives  a  difference  of  31  -f-  15,  or  2  hr.  of 
time,  with  a  remainder  of  1°.  A  difference  in  longitude  of  1°6',  or  66',  gives 
a  difference  of  66  -4-  15,  or  4  min.  of  time,  with  remainder  6'.  A  differ- 
ence of  6'  15",  or  375",  gives  a  difference  of  375  -4-  15,  or  25  sec.  of  time. 

(2)  Berlin  is  13°  23' 53"  east  longitude  and  Boston  is  71°  4' 9" 
west  longitude.  When  it  is  1.15  P.M.  at  Boston,  what  time  is  it  at 

Berlin  ? 

13°      23'         53" 
71          4  9 

15)84°       28'  2"  difference  in  longitude 

5  hr.  37  min.  52T25  sec.  difference  in  time 
1  hr.  15  min. time  in  Boston 

6  hr.  52  min.  52^  sec.  time  in  Berlin 

/.  it  is  52  min.  52T23  sec.  after  6  P.M.,  or  7  min.  7{|  sec.  to  7  P.M. 

Exercise  127 
Find  the  difference  in  time  between  the  following  cities : 

1.  Brooklyn  73°  58'  W-  and  Omaha  95°  28'  W. 

2.  St.  Paul  93°  3'  45"  W.  and  Cleveland  81°  39'  W. 


COMPOUND   QUANTITIES  189 

3.  Indianapolis  S6°6'57"  W.  and  San  Francisco  122°26'12"  W. 

4.  Cincinnati  S4°28'36"  W.  and  Glasgow  4°  17'  6"  W. 

5.  Detroit  83°  5'  7"  W.  and  Vienna  16°  22'  22"  E. 

6.  Pillsbury  79°  55'  43"  W.  and  Amsterdam  4°  52'  13"  E. 

7.  Newark  74°  9'  12"  W.  and  Rome  12°  27'  58"  E. 

8.  When  it   is  11  A.M.  at   Cleveland,  what  o'clock  is  it  at 
St.  Paul  ? 

9.  What  time  is  it  at  Indianapolis  at  the  opening  of  school 
at  9  A.M.  in  San  Francisco  ? 

10.  When  it  is  7  A.M.  at  Cincinnati,  what  time  is  it  at  Glasgow  ? 

11.  When  it  is  8  A.M.  at  Omaha,  what  time  is  it  at  Brooklyn  ? 

12.  A  man   travels  until   his  watch  is  1  hr.  5  min.  16  sec. 
slow.     Does  he  travel  east  or  west,  and  how  many  degrees  has 
he  gone  ? 

13.  A  vessel  sailed  from  a  port  directly  on  a  line  of  latitude 
a  certain  distance,  and  then  due  north  to  port,  where  the  captain 
found  that  his  chronometer  was  40  min.  slow.     In  which  direction 
did  he  sail  at  first,  and  how  many  degrees  ? 

14.  What  is  the  difference  in  longitude  between  two  places 
whose  difference  in  time  is  : 

(a)  2  hr.  33  min.  18  sec.  ? 

(b)  4  hr.  27  min.  46  sec.  ? 

(c)  6  hr.  12  min.  29  sec.  ? 

15.  Buffalo  is  78°  57' 48"  W.  and  Constantinople  is  28° 59' 3"  E. 
What  time  is  it  in  Constantinople  when  it  is  20  min.  after  6  A.M., 
July  6,  in  Buffalo  ? 

16.  What  time  is  it  in  Buffalo  when  it  is  20  min.  after  6  A.M., 
July  6,  in  Constantinople  ? 

17.  Given  the  longitude  of  two  places,  state  how  to  find  the 
time  in  the  place  east  at  a  given  time  in  the  place  west. 


190  ARITHMETIC 

Exercise  128 

1.  Find  values  of  the  quantity  measured  by  the  number  6 
according  as  the  unit  is  £2  5s.  or  6  oz.  10  pwt.  16  gr. 

2.  Find   the   value   of   a  pile   of   cordwood   13' 4"   long  by 
3' 9"  high  at  $4.50  a  cd. 

3.  How  many  gr.  are  there  in  9  oz.  17  pwt.  22  gr.,  and  how 
many  A.,  etc.,  in  167,412,715  sq.  in.  ? 

4.  In  161,384  in.  how  many  mi.  ? 

5.  If  a  sovereign  weigh  123.274  gr.,  how  many  sovereigns 
will  weigh  21  Ib.  4  oz.  16  pwt.  10  gr.  ? 

6.  The  working  wheel  of  a  locomotive  is  226  in.  in  circum- 
ference.    It  turns  91  times  in  1  min.     Through  how  many  rd., 
etc.,  does  it  draw  the  train  in  1  min.  ? 

7.  How  much  must  be  paid  for  360  ft.  of  boards  at  $  12.00 
per  M.,  250  shingles  at  $  2.50  per  M.,  and  760  ft.  of  timber  at 
$  1.00  per  C,  ? 

8.  How  many  min.  are  there  in  T5¥  yr.  -f  -gL-  wk.  +  -f%  hr.  ? 

9.  A  horse  trotted  1  mi.  in  2  min.  12  sec.     Taking  his  stride 
at  16  ft.,  how  many  times  per  sec.  did  his  feet  touch  the  ground  ? 

10.  A  bicyclist  rode  39  mi.  in  3  hr.  15  min.     What  was  his 
rate  in  mi.  per  hr.,  in  yd.  per  min.,  and  in  ft.  per  sec.  ? 

11.  Keduce  if  of  £  20  to  the  decimal  of  £  100. 

12.  How  many  oz.  of  gold  are  worth  £23  16s.  when  .53  oz. 
is  worth  44  s.  2  d.  ? 

13.  If  wire  fencing  cost  6^  per  yd.,  find  what  must   be  paid 
for  enclosing  a  field  305  yd.  long  and  156  yd.  wide,  there  being 
4  rows  of  wire. 

14.  Convert  £296   16s.   sterling  into  dollars  and  cents,  £1 
being  worth  $  4.8665. 


COMPOUND   QUANTITIES  191 

15.  Make  out  the  following  account  neatly,  accurately,  and  in 
proper  form.     All  fractions  are  to  be  retained. 

John  Wilson  bought  from  you  to-day  : 

71  Ib.  cheese  @  121^  per  Ib. ; 
61  Ib.  butter  @  23^  per  Ib. ; 
21  Ib.  tea        @  55^  per  Ib. ; 
27  Ib.  sugar    @  $  1  per  18  Ib. 

16.  If  a  yard  measure  is  1  of  an  inch  too  long,  what  is  the 
actual  distance  between  two  points  which  is  found  by  this  measure 
to  be  500  yd.  2  ft.  6  in.  ? 

17.  Express  1  Ib.  Troy  as  the  fraction  of  1  Ib.  Avoirdupois ;  and, 
conversely,  express  1  Ib.  Avoirdupois  as  the  fraction  of  1  Ib.  Troy. 

18.  Express  as  the  fraction  of  a  ft.  the  remainder  after  .012 
of  a  yd.  has  been  subtracted  as  often  as  it   is   possible   from 
1.087  yd. 

19.  Which  is  the  heavier,  a  pound  of   gold   or  a  pound  of 
feathers,  and  an  ounce  of  gold  or  an  ounce  of  feathers  ?     By  how 
much  in  each  case  ? 

20.  How  many  silver  spoons,  each  weighing  2  oz.  16  pwt., 
could  be  made  out  of  a  bar  of  silver,  the  weight  of  which  is  50  oz. 
8  pwt.  ? 

21.  If    the   unit   of    Troy   weight,    called    the    pennyweight, 
contained  14.25  gr.    instead   of  24,  how  many  gr.  would   make 
1  Ib.  Troy  ? 

22.  A  man  bought  a  quantity  of  tea  supposed  to  be  done  up  in 
packages  of  1  Ib.  each,  for  which  he  was  to  pay  $  64 ;  on  weigh- 
ing them,  however,  it  was  found  that  each  package  was  1  oz.  too 
light.     How  much  should  he  pay  for  the  tea  ? 

23.  Required  the  mean  of  the  following  observations  of  temper- 
ature :  41°  29';  41°  271';  39°  13';  41°  33';  37°  47£';  44°  28';  40°  13'. 

24.  Sold  20,900  ft.  of  lumber  for  $331.62^,  gaining  thereby 
|  78.371      What  had  it  cost  per  C.  ? 


192  ARITHMETIC 

25.  A  lot  150  ft.  long  and  100  ft.  wide  is  to  be  surrounded 
by  a  close  board  fence  6  ft.  high.    What  will  the  boards  cost  at 
$  12.50  per  thousand  ft.  ? 

26.  If  a  room  be  12  ft.  square,  what  must  its  height  be  in  order 
that  the  area  of  the  walls  may  amount  to  60  sq.  yd.  ? 

27.  Find  the  value  of  a  rectangular  field  330  yd.  by  156  yd.  @ 
$  36.50  per  A. 

28.  Find  the  surface  area  and  the  volume  of  a  rectangular 
block  3' 9"  x  2' 4"  x  1'3". 

29.  Express  as  a  fraction  of  an  A.  the  sum  of  the  following : 
i  of  4  Of  ||  of  1  A. ;  |  of  if  of  ff  of  100  sq.  rd.  ;  and  if  of  2| 
times  605  sq.  yd. 

30.  The   Manufacturers   and   Liberal   Arts    Building    of    the 
Columbian  Fair  was  in  the  form  of  a  rectangle  and  covered  an 
area  of  30  A.  76  sq.  rd.  19  sq.  yd.  7  sq.  ft.     The  building  was 
787  ft.  wide.     How  many  ft.  in  length  was  it  ? 

31.  A  200-acre  farm  is  sown  with  grain  as  follows  :  Peas,  25  A. 
126  sq.  rd.  10  sq.  yd. ;  oats,  46  A.  134  sq.  rd.  15  sq.  yd. ;  wheat, 
75  A.  125  sq.  rd.  25  sq.  yd.     The  buildings,  garden,  and  orchard 
occupy  12  A.,  and  the  rest  is  pasture.     How  many  A.  of  pasture 
are  there  ? 

32.  If  a  road  is  4  rd.  wide,  how  many  mi.  of  it  will   make 
10  A.  ? 

33.  A  map  is  drawn  to  a  scale  of  half  an  inch  to  a  mile.     How 
many  acres  are  represented  by  a  square  inch  on  the  map  ? 

34.  After  drawing  off  124  gal.  of  water  from  a  cistern,  -fT  of 
the  water  still  remained.      How  many    gal.  did   the   cistern   at 
first  contain  ?     How  many  gal.  were  left  in  it  ? 

35.  Some  Atlantic   liners   consume   200   T.   of   coal   per   day. 
They  average  8  da.  out  and  8  back.     For  fear  of  accidents  they 
carry  a  supply  for  4  da.  extra.     How  many  cu.  yd.  of  the  hold 
of  such  a  steamer  will  be  occupied  with  coal  for  her  round  trip 
if  each  ton  is  33  cu.  ft.  ? 


COMPOUND   QUANTITIES  193 

36.  A  load  of  wood  10  ft.  long,  3  ft.  8  in.  wide,  and  3  ft.  high 
was  sold  for  $  3. 

(a)  What  was  the  price  per  cd.  ? 

(b)  At  $  4  per  cd.,  what  would  the  load  be  worth  ? 

37.  Find  the  value  of  a  pile  of  tan  bark  180  ft.  long,  48  ft. 
wide,  and  16  ft.  high  at  $  2.25  per  cd. 

38.  Find   the    amount    of    the    following    bill,   retaining    all 

fractions : 

3|  lb.  tea  @  80^  ; 

300  lb.  sugar  @  4f  ^  ; 

45  yd.  print  @  11#  ; 

21  gal.  syrup  @  65^  ; 

121  yd.  towelling  @  12^  ; 

f  doz.  knives  and  forks  @  $  2.50  ; 

27  lb.  cheese  @  15^  ; 

1  lb.  10  oz.  lemon  peel  @  32  ^  per  lb. 

39.  A  train  80  yd.  long  crossed  a  bridge  140  yd.  long  in  221 
sec.     Find  the  average  speed  of  the  train  while  crossing. 

40.  Find  the  length  of  a  bridge  which  a  train  100  yd.  long 
required  1  min.  15  sec.  to  cross,  running  at  a  speed  of  15  mi.  per 
hour. 

41.  Find  the  value  of  2J  cwt.  +  37£  lb.  +  lOf  oz. 

42.  Find  the  weight  of  a  bar  3  yd.  1  ft.  9  in.  long,  of  which  a 
yard  weighs  15  lb. 

43.  Find  the  cost  price  of  lead  per  cwt.,  if  the  sale  of  48  cwt. 
for  $  218.70  gives  a  profit  of  1  of  the  original  price. 

44.  Find  the  expense  of  fencing  a  railway  (both  sides)  73  mi. 
in  length,  at  the  rate  of  $  5.50  per  rd. 

45.  If  a  wheel  makes  260  revolutions  in  passing  over  1  mi. 
520  yd.  2  ft.,  what  is  its  circumference  ? 

46.  A  block  of  stone  is  4  ft.  long,  2  ft.  6  in.  broad,  and  1  ft. 
3  in.  thick;  it  weighs  27  cwt.      Find  the  weight  of  50  cu.  in. 
of  the  stone. 


o 


194  ARITHMETIC 

47.  A  rectangular  lot  45  ft.  front  by  99  ft.  deep  was  sold  for 
$  3150.     What  was  the  price  per  ft.  frontage,  and  what  the  price 
per  A.  at  the  rate  of  the  selling  price  of  the  lot  ? 

48.  If  I  buy  147  gal.  of  molasses  at  19^  a  gal.,  and  use  33  gal. 
of  it,  at  how  much  must  I  sell  fche  remainder  per  gal.  so  as  to 
receive  as  much  as  the  whole  cost  ? 

49.  When   1  oz.   of  gold  costs  $  19.45,  what  is  the  cost  of 
.04  lb.  ? 

50.  A   bushel  of  wheat  weighs  60  lb.  and  a  barrel  of  flour 
weighs  196  lb.     If  3  lb.  of  wheat  make  2  lb.  of  flour,  how  many 
bbl.  of  flour  can  be  made  from  343  bu.  of  wheat  ? 

51.  A  grocer  receives   $  9.60  for  a  bill  of  goods  weighed  on 
scales  that  gave  only  15^  oz.  to  the  pound.     How  many  cents' 
worth  did  he  cheat  his  customer  ? 

52.  If  a  cow  gives  12  qt.  1  pt.  of  milk  every  day  and  1  lb.  8  oz. 
of  butter  can  be  made  from  25  qt.  of  milk,  how  many  pounds  of 
butter  can  be  made  in  one  week  from  the  milk  of  16  cows  ? 

53.  A  man  can  run  100  yd.  in  10  sec.     How  many  mi.  will  a 
steamboat  go  in' 5 1  da.  at  the  same  rate  ? 

54.  How  many  mi.  must  be  travelled  by  a  team  in  plough- 
ing lengthwise  a  piece  of  land  60  rd.  long  and  40  rd.  wide,  if 
each  furrow  is  10  in.  wide  ? 

55.  A  farmer  exchanges  3f  T.  of  wheat  at  64]^  per  bu.  for 
coal  at  $  6.75  per  T.     How  much  coal  does  he  get  ? 

56.  How  much  wheat  is  necessary  to  sow  a  field  containing  44 
A.,  if  f  oz.  is  sown  on  every  sq.  yd.  ? 

57.  Two  clocks  point  to  2«at  the  same  instant;   one  loses  3£ 
sec.  and  the  other  gains  4  sec.  in  12  hr.     When  will  one  be  half 
an  hour  before  the  other,  and  what  time  will   each  clock  then 
show  ? 

58.  A  railway  company  pays  $24.75  per  A.  for  a  portion  of 
road   100  mi.  long  and  94 1  ft.  wide.     Find  the  whole  amount 
paid, 


COMPOUND   QUANTITIES  195 

59.  A   man   mowing  grass  walks   at   the  rate  of   .35  mi.  an 
hour,  and  in  70  rain,  mows  a  grass  plot  of  1056  sq.  yd.     How 
broad  does  he  mow  ? 

60.  Find  the  expense  of  sodding  a  plot  of  ground,  which  is  40 
yd.  long  and  100  ft.  wide,  with  sods  each  a  yd.  in  length  and  a 
ft.  in  breadth  ;  the  sods  when  laid  costing  75^  per  hundred. 

61.  A  cd.  of  wood  and  100  bu.  of  grain  fill  equal  spaces.     A 
cubic  bin  whose  edge  is  12  ft.  contains  45,900  Ib.  of  grain.     Find 
the  weight  of  1  bu.  of  this  grain. 

62.  Make  out  the  following  bill  neatly  and  accurately.      John 
Smith,  a  merchant  of  Chicago,  sold  to  William  Jones,  on  June 
15,  1895  : 

5  Ib.  8  oz.  of  butter  @  160  per  Ib.  ; 

2  Ib.  10  oz.  of  tea  @    3^  an  oz.  ; 

4  doz.  lemons  @    4^  for  3  lemons.  ; 

8  Ib.  coffee  @  37  lj  per  Ib.  ; 

1  bu.  3  pk.  chestnuts  @  10^  per  qt.  ; 

11  doz.  penholders  @    \\$  each. 


63.  Find  cost  of  digging  a  cellar  48  ft.  long,  30  ft.  wide,  and 
6  ft.  deep,  at  20^  per  cu.  yd.,  and  flooring  it  with  Portland  cement 
at  10  ^  per  sq.  yd. 


64.  A  piece  of  land  is  surrounded  by  a  stone  wall  8  ft.  high 
and  2  ft.  thick  ;  the  land  inside  the  wall  is  100  ft.  long  and  50  ft. 
wide.     How  many  cu.  ft.  of  stone  does  the  wall  contain  ? 

65.  How  many  bushels  of  potatoes  can  be  sold  out  of  a  garden 
in  which  there  are  160  rows  of  potatoes,  in  each  row  240  hills, 
and  on  an  average  10  potatoes  in  each  hill,  if  6  potatoes  make 
3  pt.  ? 


196  ARITHMETIC 

66.    Farmer  B.  sold  to  a  merchant  the  following  articles  to 
apply  on  an  overdue  account  of  $  54.45 : 

1680    Ib.  of  hay  @  $  15  per  T. ; 

3|  cd.  of  wood  @  $  4.80  per  cd. ; 

4    bbl.  of  apples  @  $  2.75  per  bbl. ; 

350    Ib.  of  flour  @  $2.50  per  cwt. ; 

30    Ib.  10  oz.  butter  @  16^  per  Ib. 

Make  out  the  account  neatly,  showing  the  balance  and  to  whom 
due. 


CHAPTER   XIII 

PEKCENTAGE 

197.  The  expressions  $  T^7  and  $  .05  denote  that  the  quan- 
tity $1  is  conceived  as  made  up  of  100  equal  parts  or  units, 
and  that  5  of  these  parts  or  units  have  been  taken  to  measure 
the  quantity  denoted  by  T|-Q  or  .05. 

The  phrase  per  cent  means  hundredths.     Thus  the  fraction 
I^-Q  and  the  decimal  .05  are  also  written  5  per  cent  or  5%. 
Hence  yjj-g-,  .05,  and  5%  of  any  quantity  are  equal. 

198.  (1)  Express  J  as  hundredths  and  also  as  per  cent. 


100 

(2)  A  horse  dealer  who  had  600  horses  sold  480  of  them. 
What  per  cent  of  his  horses  did  he  sell  ? 

We  are  here  given  the  measured  quantity  or  480  horses  to  compare  with 
the  quantity  600  horses,  and  we  are  required  to  find  the  per  cent  which  is  the 
number. 

The  number  sold  =  f  f  §  or  f  or  80  %  of  the  whole  number. 

199.  The  term  per  cent  is  used  constantly  in  business. 
The  merchant  gains  20%,  meaning  that  he  gains  $20  on 
every  8100  he  has  invested  in  goods.  The  insurance  com- 
pany charges  2%  for  insuring  furniture,  meaning  that  §2  is 
charged  on  every  -f  100  worth  of  furniture  insured.  A  man 
borrows  money  at  5%,  meaning  that  he  is  to  pay  $5  interest 
on  every  $  100  borrowed.  The  commission  merchant  charges 

197 


198  ARITHMETIC 

2%  of  the  buying  or  selling  price.     The  broker  charges  \°/0  for 
buying  stocks,  and  so  on. 


Exercise  129 

Express  as  hundredths  and  also  as  per  cent : 

1        JL13.4:1_245_JL_3_       7_JL_3       _9        17     _1        1123 
•"••      2?    4?    4?    4'    5?    5'    5'    5?    10?    10?    10?    20?    20?    20?    20?    25?    25?    25' 

9        121513571          1          1        15      17     .14      41 
3?    3?    6?    6?    8?    8?    8?    8?    12?    15?    16?    16?    19?    27?    45* 

3.  Out  of  a  class  of  25  pupils  5  are  absent.     What  part  of  the 
class  is  absent  ?     How  many  hundredths  ?     What  per  cent  of  the 
class  ? 

4.  A  merchant  paid  $3  for  hats  which  he  sold  for  $4.     What 
fraction  of  the  cost  price  did  he  gain  ?     How  many  hundredths  ? 
What  per  cent  of  the  cost  ? 

5.  A  person  bought  a  house  for  $5000  and  afterward  sold  it 
for  $  4000.     The  loss  was  what  fraction  of  the  cost  ?     How  many 
hundredths  ?     What  per  cent  ? 

6.  A  fruit  dealer  bought  strawberries  for  $  1.75  a  crate  and  sold 
them  for  $  2.25  a  crate.     What  per  cent  did  he  gain? 

7.  A  man  bought  a  horse  for  $  234  and  afterwards  sold  it  for 
$  273.     What  per  cent  of  the  cost  did  he  gain  ? 

8.  The  population  of  a  town  of  32,000  inhabitants  increases 
1120  in  one  year.     What  is  the  per  cent  of  increase  ? 

9.  How  do  you  find  the  gain  per  cent  when  you  are  given  the 
cost  price  and  the  selling  price  of  an  article  ? 

200.   Express  25%  and  37 \%  as  fractions  in  their  lowest 
terms. 

on:  o/  _    25    _  i 
zo  /o  --  Too  -  -  ?• 

371  o/  =  §Zi  =  75  _  3^ 
100      200      8 


PERCENTAGE  199 

Exercise  130 

What  fractions  in  their  lowest   terms  are  equivalent  to  the 
following  : 


2.  60%,  75%,  35%,  24%,  70%,  16%,  85%? 

3.  121%,  371%,  621%,  87i%? 

4.  61%,  S3^%,  6|%,  16|%,  83i%? 

5.  111%,  14f  %,  9 

6.  lOf  %,  5f%,  li 

7.  100%,  120%,  125%,  175%,  250%,  325%? 

8.  A  horse  which  cost  $  120  was  sold  at  a  gain  of  25%.     The 
gain  is  equal  to  what  part  of  the  cost  ?     How  much  was  gained  ? 
What  was  the  selling  price  ? 

9.  Cloth  which  cost  60^  per  yd.  was  sold  at  a  loss  of  16f  %. 
The  loss  was  what  fraction  of  the  cost  ?     What  was  the  loss  on 
each  yd.  ?     What  was  the  selling  price  per  yd.  ? 

10.  An  article  costing  $  4.20  was  sold  at  a  gain  of  8-1%.     Find 
the  gain.     Find  the  selling  price. 

11.  Tea  is  bought  for  84^  per  Ib.  and  sold  at  an  advance  of 
14  f  %.     What  was  the  selling  price  of  each  Ib.  ? 

12.  A  drover  sold  400  sheep  at  a  gain  of  10%.     He  gained  the 
cost  price  of  how  many  sheep  ? 

13.  How  do  you  find  the  gain  on  an  article  when  you  are  given 
the  cost  price  and  the  gain  per  cent  ?     How  do  you  find  the  selling 
price  ? 

201.  The  following  results  should  be  memorized  so  that 
the  fractions  or  the  per  cent  can  be  given  rapidly  in  any 
order  : 


200  ARITHMETIC 

20%  =  i  331%  =  ^  371%  =  f  100%  =  1 

40%=  |  66|%=f  50%  -i 

60%  =f  100%=1  62i%  =| 

80%  ^f  12J#=i  75%  =  J     . 

=1  25%  =J  871%  -J 


NOTE.  —  The  expression  20%  =  ^  signifies  that  20%  of  a  quantity  =  |  of 
it.     Thus,  20  %  of  80  A.  =  I  of  80  A. 


Exercise  131 

Read  the  following  decimals  as  per  cents  : 

1.  .25,  .16-|,  .40,  .75,  .031,  1.20. 

2.  .15,  .371,  .45£,  .051  .001  2.40. 

202.   What  is  the  quantity  which  is  measured  by  the  unit 
1840  and  the  number  16f  %  ? 

o  ' 

The  quantity  =  16  f  %  of  $  840  =  £  of  $  840  =  $  140. 

Exercise  132 

1.  What  is  the  quantity  which  is  measured  by  the  unit  $720 
and  the  number  8i%? 

2.  What  is  the  quantity  which  is  measured  by  the  unit  $465 
and  the  number  20%? 

3.  What  is  the  gain  that  is  measured  by  the  cost  $893  and 
the  number  6%? 

4.  What  is  the  loss  that  is  measured  by  the  cost  $1268  arid 
the  ratio  32%? 

5.  What  is  the  quantity  that  is  measured  by  the  sum  of  the 
unit  $  468.25  and  8%  of  it  ? 

6.  What  is  the  quantity  that  is  measured  by  the  difference 
between  the  unit  $4397.50  and  6%  of  it? 


PERCENTAGE  201 

7.  If  the  gain  on  selling  is  measured  by  the  number  7%  and 
the  cost  $  450,  find  the  gain. 

8.  Find  the  selling  price  of  an  article  which  cost  $  600,  the 
loss  on  selling  being  measured  by  the  number  5%  and  the  cost 
price. 

203.  A  speculator  bought  a  house  for  $  2349  and  sold  it 
at  a  gain  of  17%.  Find  the  selling  price. 

In  this  question  the  selling  price  is  the  sum  of  the  cost,  which  is  known, 
and  the  gain,  which  is  unknown.  The  gain  is  measured  by  the  number  17  % 
or  .17,  and  the  cost  price  $2349. 

$2349  cost 

.17  number 
16443 
2349 


$399.33  gain 

The  gain  ==  17%  of  $2349  =  $399.33. 
.-.  the  selling  price    =  $2349  +  $399.33  =  $2748.33. 

Exercise  133 

1.  Write  as  decimals:  17%,  13%,  37%,  231%,  146%,  346%, 

6%,  8%,  81%,  li%,  1%,  1%. 

2.  Find  34%  of  $893. 

3.  Find  27%  of  6594  bu.  of  wheat. 

4.  If  39%   of  a  cargo  of  flour,  consisting  of  8492  bbl.,  was 
damaged,  how  many  bbl.  were  damaged  ? 

5.  A  farmer  who  sold  his  crop  of  wheat  in  1892  for  $  967.20, 
received  13%  less  the  next  year.     How  much  less  did  he  receive 

for  his  crop  in  1893  than  in  1892  ? 

•^ 

6.  A  grain  dealer  invested  $6459  in  wheat,  and  23%  of  that 

amount  in  oats.     How  much  did  he  invest  in  oats  ? 

7.  What  does  a  bill  for  $  1896  become  after  a  reduction  of  3% ? 


202  ARITHMETIC 

8.  What  is  the  selling  price  of  an  article  costing  $  18,  and 
sold  at  a  loss  of  9%? 

9.  What  is  the  selling  price  of   an  article  costing  $  1,  and 
sold  at  a  gain  of  7%? 

204.   Express  -|%  as  a  decimal  and  also  as  a  common  frac- 
tion in  its  lowest  terms. 


100      500      125 

/.  fo/  =.004=    i 

O     / '.'  O  ±,  ft  O 

Exercise  134 

Express  as  decimals  and  also  as  common  fractions  in  their 
lowest  terms : 


f  %,  !%,  f 


2-  f  %,  f  %,  i%,  |%,  f  %,  i%,  f  %,  T4T%- 

3-  A%,  T?0%.  A%,  M%>  T9T%?  lt%?  T 

4.  What  part  of  1%  is  £%?  f%?  |%?  T7¥%? 

5.  If  ^-%  is  charged  for  sending  money  from  Chicago  to  New 
York,  what  is  charged  for  sending  $  1200  ? 

6.  If  -J-%  is  charged  for  sending  money  to  St.  Louis,  find  how 
much  is  charged  when  $  1632  is  sent. 

7.  If  \%  is  charged  as  commission  for  buying  stock,  what  is 
the  commission  on  buying  $  2400  stock  ? 

8.  If  -|-%  is  charged  for  selling  stock,  find  the  commission 
charged  for  selling  $  1600  stock. 

9.  6%  per  annum  is  what  per  cent  for  1  month  ?  -f  %  a  month 
is  what  per  cent  per  annum  ? 

10.  What  is  the  cost  of  insuring  550  bbl.  of  flour,  worth 
$  4  per  bbl.,  the  cost  of  insurance  being  ^-%  of  the  value  of  the 
flour? 


PERCENTAGE  203 


205.    (1)  Express  102-*  %  as  a  decimal. 


1021%  =        =  1.021  =  1.025. 

100 

(2)  I  send  my  agent  $5100,  which  is  102%  of  the  money 
which  he  invested  for  me  in  cotton.  What  does  my  agent 
pay  for  cotton? 

102%  or  1.02  of  the  cost  of  the  cotton  =  $5100. 
.-.  the  cost  of  the  cotton  =  $5100  -f-  1.02  =  $  5000. 

Exercise  135 

1.  Express  as  decimals:  103%,  104%,  101%,  102%. 

2.  Express  as  decimals:  97%,  96%,  99%,  98%. 

3.  Express  as  decimals  :  103^%,  1041%,  102f%,  1011%. 

4.  Express  as  decimals:  971%,  96f%,  99J%,  98f%. 

5.  A  real  estate  broker  in  St.  Louis  received  $  5047,  which  was 
103%  of  the  sum  he  was  to  invest  in  real  estate.     What  sum  did 
he  invest  in  real  estate  ? 

6.  My  agent  sent  me  $3395,  which  was  97%  of  the  selling 
price  of  some  railway  stock.     What  did  the  stock  sell  for  ? 

7.  My  agent  on  selling  a  quantity  of  wheat  sent  me  98^%  of 
the  proceeds.     If  I  received  $  3546,  what  did  the  wheat  sell  for  ? 

8.  I  sent  my  agent  $4545.30,  which  was  104^%  of  the  sum 
he  was  to  invest  in  buying  silk.     What  did  he  pay  for  the  silk  ? 

206.  (1)  A  house  is  sold  for  $16,400,  and  25%  of  the 
purchase  money  is  paid  down,  the  balance  to  remain  on  mort- 
gage. How  much  remains  on  mortgage  ? 

In  this  problem  we  are  given  the  measured  whole,  i.e.  the  selling  price, 
and  the  number  or  25  %  of  it.  We  are  required  to  find  the  balance  which  is 
the  difference  between  the  selling  price  and  the  sum  paid. 

The  sum  paid  =  25%  or  \  of  $  16,400  ==  .$4100. 

.-.  the  balance  =  $16,400  -  $4100  =  $  12,300. 


204  ARITHMETIC 

(2)  On  Jan.  10  a  merchant  buys  goods  invoiced  at  $  876.40, 
on  the  following  terms:  4  months,  or  less  6%  if  paid  within 
10  days.  What  sum  will  pay  the  debt  on  Jan.  15? 

Since  Jan.  15  is  less  than  10  days  after  Jan.  10,  the  sum  due  will  be  6% 
less  than  $876.40. 

The  discount  =  6  %  or  .06  of  $876.40  =  $52.584. 
.-.  the  sum  due  =  $876.40  -  $52.58  =  $823.82. 

Exercise  136 

1.  A  maltster  malts  7500  bu.  of  barley,  which  in  the  process 
increases  124  %.     How  many  bu.  of  malt  has  he  ? 

2.  Certain  books  are  bought  at  $  1.75  each.     At  what  must 
they  be  sold  to  gain  12%  ? 

3.  A  merchant  asked  30%  advance  on  goods  which  cost  $120, 
but  finally  took  25%  less  than  the  price  asked.    What  did  he  sell 
them  for  ? 

4.  A  merchant  bought  apples  at  60^  per  bu.,  and  sold  them 
at  a  gain  of  25%.     Find  the  selling  price  per  bu.    How  many 
bu.  did  he  sell  if  he  received  all  together  $37.50  ? 

5.  Bought  $  64  worth  of  apples  at  80^  per  bu.,  part  of  which 
being  damaged  and   rendered   worthless,   I   sold   the  remainder 
at  an  advance  of  50%,  receiving  $76.80.     How  many  bu.  were 
damaged  ? 

6.  If  10%  of  an  army  of  23,400  men  were  slain  in  battle,  and 
5  per  cent  of  the   remainder  were   mortally  wounded,  find  the 
sum  of  the  killed  and  mortally  wounded. 

7.  The  population  of  a  town  of  64,000  inhabitants  increases  at 
the  rate  of   2|-%  in  each  year.     Find  its  population  1,  2,  and 
3  years  hence. 

8.  The  population  of  a  city  is  a  million;  it  increases  liJ-%  for 
3  years  successively.     Find  the  population  at  the  end  of  3  years. 


PERCENTAGE  205 

9.  A  lawyer  collected  $287.50,  and  charged  5%  for  his  ser- 
vices. How  much  did  he  retain  and  how  much  did  he  pay 
over  ?  What  per  cent  is  the  amount  paid  over  of  the  amount 
collected  ? 

10.  The  cost  price  of  a  book  is  $  1.60,  the  expense  of  sale  5% 
upon  the  cost  price,  and  the  profit  25%  upon  the  whole  outlay. 
Find  the  selling  price  of  the  book. 

11.  The  cattle  on  a  certain  stock  farm  increase  at  the  rate  of 
18f  %  per  annum.     If  there  are  4096  cattle  in  1894,  how  many 
will  there  be  in  1896  ? 

12.  A  man  bought  a  store  and  contents  for  $  4720 ;  he  sold  the 
same  for  12.}%  less  than  he  gave,  and  then  lost  15%  of  the  sell- 
ing price  in  bad  debts.     Find  his  entire  loss. 

13.  A  person  having  bought  goods  for  $40  sells  half  of  them 
at  a  gain  of  5%.     For  how  much  must  he  sell  the  remainder  so 
as  to  gain  20  %  on  the  whole  ? 

14.  A  grocer  mixes  two  kinds  of  tea  which  cost  him  38^  and 
44^  per  Ib.  respectively.    What  must  be  the  selling  price  of  the 
mixture  in  order  that  he  may  gain  15  %  on  his  outlay  ? 

15.  A  grain  dealer  expended  $  2150  in  the  purchase  of  wheat, 
one-half  as  much  again  in  the  purchase  of  barley,  and  twice  as 
much  in  the  purchase  of  corn ;  he  sold  the  wheat  at  a  profit  of 
6%,  the  barley  at  a  loss  of  5%,  and  the  corn  at  a  gain  of  2%. 
Find  his  gain  on  the  whole  transaction. 

16.  A  person  gave  $  150  for  one  horse  and  $  225  for  another. 
He  sold  the  first  horse  at  a  gain  of  20%,  and  the  second  at  a  loss 
of  20%.    Find  the  selling  price  of  each  horse  and  the  gajn  or  loss 
on  the  whole  transaction. 

17.  A  sells  goods  to  B  which  cost  him  $465,  at  a  gain  of  6%, 
B  sells  them  to  C  at  a  loss  of  3%,  and  C  sells  them  to  D,  gaining 
10%.     What  did  D  give  for  the  goods  ? 


206  ARITHMETIC 

18.  A  man  having  bought  a  lot  of  goods  for  $450  sells  1  at 
a  loss  of  5%,  i  at  a  gain  of  7%,  and  the  remainder  at  a  gain  of 
2%.     Find  the  total  gain. 

19.  A  merchant  began  business  with  a  capital  of  $  30,000.     He 
gained  16f  %  the  first  year,  which  he  added  to  his  capital,  and 
121%  the  second  year,  which  he  added  to  his  capital.     In  the 
third  year  he  lost  20%.     Find  his  capital  at  the  end  of  the  third 
year. 

20.  Sugar  being  composed  of  49.856%  of  oxygen,  43.265%  of 
carbon,    and    the    remainder   hydrogen,    find   how   many   Ib.    of 
each  of  these  materials  there  are  in  1  T.  of  sugar. 

21.  A  merchant  buys  a  bill  of  dry  goods,  April  16,  amounting 
to  $6377.84,  on  the  following  terms:  4  mo.,  or  less  5%  if  paid 
within  30  da.     How  much  would  settle  the  account  on  May  16  ? 
The  amount  paid  May  16  is  what  per  cent  of  the  full  amount 
of  the  bill  ? 

22.  Water   is    composed   of   88.9%   of   oxygen  and  11.1%   of 
hydrogen.     How  many  Ib.  are  there  of  each  in  1  cu.  yd.  of  water  ? 
(A  cu.  ft.  of  water  weighs  1000  oz.) 

207.    If  a  gain  of  $  24  is  measured  by  a  certain  number  and 
the  cost  price,  which  is  $  64,  find  the  number  as  a  rate  per 

cent. 

The  required  number  x  $  64  =  $  24. 
.-.  the  required  number  =  $||  =  ff  =  f  =  371%. 

Exercise  137 

1.  If, the  quantity  of  land  sold,  which  is  50  A.,  is  measured 
by  a  certain  number  and  a  unit  of  200  A.,  find  the  number  as  a 
rate  per  cent. 

2.  If  a  gain  of  $45  is  measured  by  a  certain  number  and  the 
cost  $  225,  find  the  number  as  a  rate  per  cent. 


PERCENTAGE  207 

3.  If  a  loss  $36  is  measured  by  a  certain  rate  per  cent  and  the 
cost,  which,  is  $  108,  find  the  rate  per  cent. 

4.  If  the  sum  of  money  paid  for  fruit,  which  is  $  6,  is  measured 
by  a  certain  number  and  the  unit  $  72,  find  the  number  as  a  rate 
per  cent. 

208.    (1)  A  merchant  sold  60  yd.  of  cloth  from  a  web  con- 
taining 150  yd.     What  per  cent  of  the  web  did  he  sell  ? 

"VYe  are  here  given  the  measured  part  and  the  measured  whole,  and  we  are 
required  to  find  the  number  expressing  the  ratio  of  the  part  to  the  whole. 
The  quantity  sold  =  T6-^  or  f  or  40  %  of  the  web. 

(2)  An  article  which  cost  I  3.60  was  sold  for  $4.32.    Find 
the  gain  per  cent. 

The  gain  =  $4.32  --  $3.60  =  72  cents. 
.•.  the  gain  =  -g7^  or  |  or  20  %  of  the  cost. 

Exercise  138 

1.  The  cost  price  of  an  article  is  $64,  and  the  gain  on  selling 
$  16.     Find  the  gain  per  cent. 

2.  Tea  is  bought  at  84^  per  lb.,  and  sold  at  98^.     Find  gain 
per  cent. 

3.  Out  of  48  eggs,  6  were  broken.     What  per  cent  of   the 
whole  number  was  broken  ? 

4.  The  cost  price  of  an  article  was  $  56,  and  the  selling  price 
$  49.     Find  the  loss  per  cent. 

5.  What  per  cent  is  1  in.  of  1  ft.  ?     1  ft.  of  1  yd.  ?     1  yd.  of 
1  rd.  ?     1  rcl.  of  1  mi.  ? 

6.  What  per  cent  is  1  min.  of  1  hr.  ?     1  hr.  of  1  da.  ?     1  da. 
of  1  wk.  ?     1  wk.  of  1  yr.  ? 

7.  What  per  cent  is  1  pt.  of  1  qt.  ?     1  qt.  of  1  gal.  ? 

8.  What  per  cent  of  1  pk.  is  1  qt.  ?    Of  1  bu.  is  1  pk.  ? 


208  ARITHMETIC 

• 

9.    A  merchant  by  selling  1  Ib.  of  butter  gains  the  cost  price 
of  1  oz.     What  is  his  gain  per  cent  ? 

10.  One  Ib.  Troy  is  what  per  cent  of  1  Ib.  Avoir.  ?     One  Ib. 
Avoir,  is  what  per  cent  of  1  Ib.  Troy  ? 

11.  The  volume  of  1  gal.  is  what  per  cent  of  1  cu.  ft.?     A 
cu.  ft.  is  what  per  cent  of  1  gal.  ? 

12.  An  area  containing  1  sq.  yd.  is  increased  by  4  sq.  ft.    Find 
the  per  cent  of  increase. 

13.  If  £  1  is  worth  $  4.866,  what  per  cent  of  £  1  is  $  1  ? 

Exercise  139 

1.  A  paymaster  receives   $  150,000   from   the   treasury,  but 
fails  to  account  for  $  2250.    What  is  the  percentage  of  loss  to  the 
government  ? 

2.  A  city  of  17,000  inhabitants  increases  in  a  given  time  to 
20,000.     Find  the  increase  per  cent. 

3.  $  640  increased  by  a  certain  per  cent  of  itself  equals  $  720. 
Required  the  rate  per  cent. 

4.  A  house  worth  $  3500  rents  for  $  400.     For  what  per  cent 
of  its  value  does  it  rent  ? 

5.  If  a  tradesman  gain  $  1.32  on  an  article  which  he  sells  for 
$  5.28,  what  is  his  gain  per  cent  ? 

6.  An  article  which  cost  84^  is  sold  for  93^.     Find  the  gain 
per  cent. 

7.  A  city  gained  2467  in  population  in  5  years.     If  its  popu- 
lation was  14,802  five  years  ago,  what  was  the  gain  per  cent  ? 

8.  In  a  certain  year  the  number  of  graduates  of  a  school  was 
70.      Ten  years  later  it  was  213.      Find  the  rate  per  cent  of 
increase. 

9.  A  tea  merchant  mixes  40  Ib.  of  tea  at  45^  per  Ib.  with 
50  Ib.  at  27^  per  Ib.,  and  sells  the  mixture  at  42^  per  Ib.     What 
per  cent  profit  does  he  make  ? 


PERCENTAGE  209 

10.  Paid  $  664.25  for  transportation  on  an  invoice  of   goods 
amounting  to  $  8866.    What  per  cent  must  be  added  to  the  invoice 
price  to  make  a  profit  of  20  %  on  the  full  cost  ? 

11.  A  grocer  uses  for  a  1  Ib.  weight  one  which  only  weighs 
15.75  oz.     What  does  he  gain  per  cent  by  his  dishonesty  ? 

12.  I  bought  500  sheep  at  $  6  a  head ;  their  food  cost  me  $  1.25 
a  head ;  I  then  sold  them  at  $  10  a  head.     Find  my  whole  gain, 
and  also  my  gain  per  cent. 

13.  A  man's  income  is  derived  from  the  proceeds  of  $  4550  at 
a  certain  rate  per  cent ;  and  $  5420  at  1  %  more  than  the  former 
rate.     His  whole  income  being  $  453,  find  the  rates. 

14.  If  a  commodity  be  bought  for  $  16.42  a  cwt.  and  sold  for 
18^  a  Ib.,  find  the  rate  of  profit  per  cent. 

15.  The  police  returns-  for   a   certain   year   give   1350   male 
offenders,    and   1150   female;    the   next   year's   returns    show   a 
decrease  of  6%  in  the  number  of  male  criminals,  and  an  increase 
of  8%  in  number  of  female.     Find  increase  or  decrease  per  cent 
in  whole  number  of  criminals. 

16.  The  area  of  North  America  is  8,000,000  sq.  mi.,  and  of  the 
Mississippi  and  St.  Lawrence  basins  respectively  1,250,000  and 
350,000  sq.  mi.     Find  what  per  cent  of  the  area  of  the  continent 
is  drained  by  each  of  these  rivers. 

17.  Turn  to  your  geography  and  find  the  area  of  the  basins  of 
the  principal  rivers,  and  then  find  what  per  cent  of  the  area  of 
North   America  is  drained  by  each  river  and  by  all  of  them 
together. 

18.  The  length  of  the  Missouri-Mississippi  Eiver  is  4200  mi., 
and  of  the  St.  Lawrence  2000  mi.      The  length  of  the  St.  Law- 
rence is  what  per  cent  of  that  of  the  Missouri-Mississippi  ?     The 
area  of  the  St.  Lawrence  basin  is  what  per  cent  of  the  area  of  the 
Mississippi  basin  ? 


210  ARITHMETIC 

209.    If  a  gain  of  836  is  measured  by  the  number  22|%  and 
the  cost,  find  the  cost. 

The  gain  =  22  f  %  or  f  of  the  cost. 
|  of  the  cost  =  $36. 
^  of  the  cost  =  $  18. 
f  of  the  cost  =  $  162. 
.-.  the  cost  =  $162. 

Exercise  140 

1.  If  the  quantity  $18  is  measured  by  the  number  6%  and 
a  certain  unit,  find  the  unit. 

2.  If  the  gain  $32  is  measured  by  the  number  8%   and  the 
cost,  find  the  cost. 

3.  If  the  loss  $60  is  measured  by  the  ratio  12%  and  the  cost, 
find  the  cost. 

4.  If   the    selling   price    $3810  is   measured   by  the  number 
127%  and  the  cost  price,  find  the  cost  price. 

5.  If  the  selling  price  $5640  is  measured  by  the  number  94% 
and  the  cost,  find  the  cost. 

210.    A  trader  sold  a  horse  at  an  advance  of  12%,  gaining 
$18.     Find  the  cost  price  of  the  horse. 

12  %  or  235  of  the  cost  =  $  18. 
^  of  the  cost  =  $  6. 
|f  of  the  cost  =  $  150. 
.•.  the  cost  =  $  150. 

Exercise  141 

1.  A  quantity  of  sugar  was  sold  at  an  advance  of  121%.     If 
the  gain  was  $  17,  what  was  the  cost  ? 

2.  A  yd.  of  cloth  was  sold  at  a  loss  of  374-%.     If  the  loss  was 
36  f  a  yd.,  what  was  the  cost  ? 


PERCENTAGE  211 

3.  Forty-five  per  cent  of  a  piece  of  cloth  was  sold.     If  135 
yd.  were  sold,  how  many  yd.  were  in  the  piece  at  first  ?     How 
many  remain  unsold?     What  per  cent  remains  unsold  ? 

4.  If  3%  more  be  gained  by  selling  a  horse  for  $333  than  by 
selling  him  for  f  324,  find  his  original  price. 

5.  A  man  bought  a  horse  which  he   sold  at  a  loss  of   8%. 
If  he  had  received  $24  more,  he  would  have  gained  7%.     What 
did  the  horse  cost  him  ? 

6.  A  clerk  pays  16%  of  his  salary  each  year  for  board.    If  his 
board  costs  him  $  208  a  year,  what  is  his  salary  ? 

7.  A  man  sold  a  field  consisting  of  15  A.,  which  was  6J%  of 
his  farm.     How  many  A.  were  in  the  farm  at  first  ? 

8.  If  25%  of  my  money  is  invested  in  bank  stock  and  the 
remainder  in  business,  what  per  cent  of  my  money  is  invested 
in  business  ?     My  bank  stock  is  worth  what  part  of  my  business 
capital  ?     If  I  have  $  4800  invested  in  business,  what  is  the  value 
of  my  bank  stock  ? 

9.  Twenty  per  cent  of  my  money  is  invested  in  business,  and 
the  remainder,  which  is  $  12,800,  in  real  estate.     How  much  have 
I  invested  in  business  ? 

10.  I  invested  25%  of  my  money  in  business,  and  put  6|%  of 
the  remainder  in  the  bank.     If  I  put  $  600  in  the  bank,  how 
much  money  did  I  have  at  first  ? 

11.  Twenty-eight  per  cent  of  a  sum  of  money  was  invested  in 
business,  and  121%  of  the  remainder  in  real  estate.     If  the  sum 
invested  in  business  exceeds  that  invested  in  real  estate  by  $  1900, 
find  the  amount  of  money  I  had  at  first. 


211.   (1)  A  man  invests  77  J%  of  his  capital  in  bank  stock, 
and  has  $  29,367  left.     What  is  bis  capital  ? 

The  amount  left  =  100%  -  77|%  or  22  1-%  of  his  capital. 
22p/0  or  _9_  Of  his  capital  =  $29,367. 
•fa  of  his  capital  =  $  3263. 
f  §  of  his  capital  =  $  130,520. 
.-.  his  capital  =  $  130,520. 


212  ARITHMETIC 

(2)  If  a  profit  of  17%  is  made  by  selling  an  article  at  an 
advance  of  $24.50,  what  would  have  been  the  selling  price  if 
the  loss  had  been  8%  ? 

17%  of  the  cost  =  $24.50. 
1%  of  the  cost  =  $1.4412. 
100%  of  the  cost  =  $  144.12. 
The  loss  =  8%  of  the  cost  =  $  11.53. 
.-.  the  second  selling  price  =  $  144.12  -  $  11.53  =  $  132.59. 

Exercise  142 

1.  A  man  invests  42%  of  his  capital  in  real  estate,  and  has 
$  53,070  left.     What  is  his  capital  ? 

2.  A  bankrupt's  assets  are  $23,625,  and  he  pays  40%  of  his 
liabilities.     What  are  his  liabilities  ? 

3.  A  merchant   loses  6J%   of   the  cost   price   by  selling   an 
article  at  a  loss  of  $  27.30.     Find  the  cost  price,  and  also  at  what 
he  must  sell  it  to  gain  7J%. 

4.  By  selling  a  house  at  a  loss  of  $  150,  a  real  estate  dealer 
loses  6|%  of  the  cost.     Find  the  cost  and  also  the  gain  per  cent 
if  it  had  been  sold  for  $  2625. 

5.  I    sold  a  lot   at   a   gain   of  8J%,  thereby  gaming   $113. 
What  should  I  have  sold  it  for  to  gain  9%? 

6.  Coals  are  20%  cheaper  this  year  than  last.     If  the  price 
were  to  rise  $  1  per  T.,  they  would  still  be  50^  per  T.  cheaper 
than  last  year.     Find  the  price  last  year. 

7.  A  person  asked  for  a  lot  of  land  40%  more  than  it  cost  him, 
but  finally  reduced  his  price  15%,  gaining  on  the  whole  $1000. 
For  how  much  did  he  sell  the  land  ? 

8.  A  merchant  sold  f  of  a  quantity  of  cloth  at  a  gain  of  20%, 
and  the  remainder  at  cost.     His  gain  was  what  per  cent  of  the 
cost  ?     If  he  gained  $  7.29,  what  was  the  cost  of  the  goods  ? 

9.  A  merchant  sold  f  of  a  quantity  of  tea  at  a  gain  of  12%, 
and  the  remainder  at  a  gain  of  9%,  gaining  all  together  $2.75. 
Find  the  cost  of  the  tea. 


PERCENTAGE  213 

10.  A  speculator  gained  20%  on  f  of  his  investment,  and  lost 
24%  on  the  remainder.     All  together  he  made  $  270.     Find  the 
amount  of  his  investment. 

11.  A.  business  firm's  resources  consist  of  notes,  merchandise, 
personal  accounts,  etc.,  to  the  amount  of  $  9117.61,  and  a  balance, 
which  is  44%  of  their  entire  capital  on  deposit  in.  bank.     How 
much  is  on  deposit  ? 

12.  Ten  per  cent  of  an  army  were  slain  on  the  field  of  battle, 
and  5  per  cent  of  the  remainder  were  mortally  wounded.     The 
difference  between  the  killed  and  mortally  wounded  was  1100. 
How  many  men  went  into  battle  ? 

212.    (1)   A  horse  was  sold  for  $117,  which  was  8^%  more 
than  it  cost.     Find  the  cost  price. 

The  gain  =  8|  %  or  T\  of  the  cost. 
The  selling  price  =  if  of  the  cost, 
if  of  the  cost  =  $117. 
&  of  the  cost  =  $  9. 
if  of  the  cost  =  $  108. 
.-.  the  cost  =  $  108. 

(2)  A  horse  was  sold  for  f  154,  which  was  8J%  less  than 
it  cost.     Find  the  cost  price. 

The  loss  =  8i%,  or  ^  of  the  cost. 
The  selling  price  =  i|  of  the  cost, 
ii  of  the  cost  =  $  154. 
T\  of  the  cost  =  $  14. 
\l  of  the  cost  =  $  168. 
.-.  the  cost  =  $  168. 

Exercise  143 

1.  A  house  and  lot  were  sold  for  $  3600,  which  was  20%  more 
than  they  cost.     Find  the  cost  price. 

2.  A  house  and  lot  were  sold  for  $4200,  which  was  25%  less 
than  they  cost.     Find  the  cost  price. 


214  ARITHMETIC 

3.  A  speculator  gained  7%  by  selling  wheat  for  $2140.     Find 
the  cost  price. 

4.  Eggs  are  sold  at  the  rate  of  15^  per  doz.,  a  profit  of  25% 
being  made.     What  is  the  cost  price  per  doz.  ? 

5.  I  sold  a  book  for  42^,  gaining  16f  %.     Find  the  cost  price. 
How  much  would  be  gained  by  selling  at  a  gain  of  25%?     What 
would  then  be  the  selling  price  ? 

6.  By  selling   hats   at   60^   each,  a  merchant   gains  33|%. 
Find  the  cost  price.     What  would  have  been  the  actual  loss,  and 
what  the  loss  per  cent,  if  they  had  been  sold  at  36  ^  each  ? 

7.  I  sold  a  lot  of  land  for  $  600,  thereby  gaining  20%.    Find 
the  cost  price. 

8.  I  sold  a  lot  of  land  for  $600,  thereby  losing  20%.     Find 
the  cost  price. 

9.  What  is  the  cost  of  both  lots  in  questions  7  and  8  ?    What 
is  their  selling  price  ?    How  much  is  the  cost  of  both  greater  than 
their  selling  price  ? 

10.  A  dealer  sold  two  bicycles  for  $45  each,  losing  25%  on  one 
and  gaining  25%  on  the  other.  How  much  did  he  lose  on  the 
whole  transaction  ? 

213.  (1)  If  a  debt,  after  a  reduction  of  3%,  becomes 
$1008.80,  what  would  it  become  after  a  reduction  of  4%  ? 

After  a  reduction  of  3  %,  the  amount  owed  =  97  %  of  the  original  debt,  and 
after  a  reduction  of  4  %  it  becomes  96  %  of  the  original  debt. 

97%  of  the  debt  =  $1008.80. 

1  %  of  the  debt  =  $  10.40. 
96%  of  the  debt  =  |998.40. 
.  •.  after  a  reduction  of  4  %  the  debt  =  $  998.40. 

(2)  The  population  of  a  city  increases  2%  yearly.  It  now 
has  132,651  inhabitants.  How  many  had  it  1,  2,  and  3  years 
ago? 


PERCENTAGE  215 

The  population  now  =  102%  or  1.02  of  that  1  year  ago. 

1.02  of  the  population  1  year  ago  =  132,651. 

.-.  the  population  1  year  ago  =  132,651  -s-  1.02  =  130,050. 
.  •.  the  population  2  years  ago  =  130,050  -*-  1.02  =  127,500. 
.-.  the  population  3  years  ago  =  127,500  +  1.02  =  125,000. 

Prove  these  answers  correct. 

Exercise  144 

1.  A  horse  was  sold  for  $658,  which  was  16f  %  more  than  its 
cost.     How  much  did  it  cost  ? 

2.  A  speculator  gained  3%  by  selling  wheat  for  $6437.50. 
Find  the  cost  price. 

3.  A  merchant,  after  a  business  of  5  years,  found  his  capital 
increased  to  $28,000,  showing  a  gain  of  60%  on  his  original  capi- 
tal.    Find  that  capital. 

4.  Eggs  are  sold  at  the  rate  of  5  for  6^,  a   profit  of  20% 
being  made.     Find  the  price  at  which  they  are  bought. 

5.  In  1896  a  city  has  a  population  of  28,000  inhabitants.     If 
its  population  increased  \1\\°/0  in  the  two  years  previous,  what 
was  it  in  1894  ?     If  its  population  decreased  17-j-4%  in  the  two 
years  previous,  what  was  it  in  1894  ? 

6.  By  selling  an  article   for   $2.64  a  merchant  loses  12%. 
What  was  the  cost,  and  for  what  must  he  sell  it  to  gain  16|%? 

7.  A  merchant  sells  tea  at  75^  per  lb.,  thereby  losing  5%. 
What  was  the  cost,  and  at  what  price  per  lb.  must  it  be  sold  to 
gain  4i%? 

8.  Flour  is  sold  for  $6  per  bbl.,  at  a  loss  of  17%.      Find 
what  selling  price  would  give  16%. 

9.  If,  by  selling  an  article  for  $25.30,  8%  be  lost,  what  per 
cent  is  gained  or  lost  if  it  be  sold  for  $  38  ? 

10.    I  sold  goods  at  $21.60  per  cwt.,  thereby  gaining  14|-%. 
Find  the  cost  per  lb. 


216  ARITHMETIC 

11.  A  farmer  sold  his  crop  of  wheat  in  1871  for  8%  more  than 
he  obtained  for  his  crop  of  the  preceding  year ;  he  received  for 
both  crops  $2600;  how  much  did  he  get  for  his  crop  of  1870? 

12.  I  sold  two  houses,  receiving  $  2400  for  each.     On  the  first 
I  gained  25%,  and  on  the  second  lost  25%.     Find  the  actual  loss 
and  also  the  loss  per  cent. 

13.  By  selling  a  lot  of  land  for  $600,  thereby  gaining  20%  ;  a 
second  for  $  600,  losing  20%  ;  and  a  third  at  an  advance  of  20% 
on  cost ;  I  find  I  have  made  $  75  on  the  whole  transaction.     Find 
the  cost  of  each  lot. 

14.  A.  ship  depreciates  in  value  each  year  at  the  rate  of  10% 
of  its  value  at  the  beginning  of  the  year,  and  its  value  at  the  end 
of  three  years  is  $  14,580.     What  was  its  original  value  ? 

15.  The  cattle  on  a  stock  farm  increase  at  the  rate  of  18J% 
per  annum.     In  1889  there  were  6859  head  of  cattle  on  the  farm ; 
how  many  were  there  in  1888  ?     In  1887  ?     In  1886  ? 

16.  An  importer  pays  for  freight  and  duty  10%  on  cost  price, 
and  sells  to  the  retailer  at  a  profit  of  20%  ;  the  retailer  sells  to 
the  consumer  at  a  profit  of  25%.     Find  the  amount  paid  by  the 
consumer  for  goods  which  cost  the  importer  $  7500. 


PROFIT  AND  Loss 

214.  The  Profit  is  the  amount  by  which  the  selling  price 
exceeds  the  buying  price. 

The  rate  of  profit  is  usually  expressed  as  a  certain  per  cent 
of  the  cost  price. 

The  Loss  is  the  amount  by  which  the  selling  price  falls 
short  of  the  cost  price. 

The  rate  of  loss  is  usually  expressed  as  a  certain  per  cent 
of  the  cost  price. 


PERCENTAGE  217 

215.  (1)  At  a  forced  sale  a  bankrupt's  house  was  sold  for 
18000,  which  was  20%  less  than  its  real  value.  If  the  house 
had  been  sold  for  >£  12,000,  what  per  cent  of  its  real  value 
would  it  have  brought? 

80  %  of  the  value  of  the  house  =  $    8000. 
1  %  of  the  value  of  the  house  =         100. 
100  %  of  the  value  of  the  house  =    10,000. 
.-.  the  second  selling  price  =  %%%%$  of  the  value 

=  120  %  of  the  value. 

(2)  The  manufacturer  of  an  article  makes  a  profit  of  25%, 
the  wholesale  dealer  makes  a  profit  of  20%,  and  the  retail 
dealer  makes  a  profit  of  30%.  What  is  the  cost  to  the  manu- 
facturer of  an  article  that  retails  at  $15.60? 

Let  the  cost  to  the  manufacturer  =  100  units  of  money. 

The  selling  price  of  the  manufacturer         :  125  units  of  money. 
The  gain  of  the  wholesale  dealer  =    25  units  of  money. 

The  selling  price  of  the  wholesale  dealer  =  150  units  of  money. 
The  gain  of  the  retail  dealer  =    45  units  of  money. 

The  selling  price  of  the  retail  dealer         =195  units  of  money. 

195  units  of  money  =  $15.60. 

1  unit    of  money  =         .08. 
100  units  of  money  =       8.00. 

.-.  the  prime  cost  =       8.00. 


GENERAL  STATEMENT  OF  SOLUTION 

(3)  Represent  the  cost  to  the  manufacturer  by  100  units 
of  money,  and  then  find  the  number  of  units  representing 
respectively  the  selling  prices  of  the  manufacturer,  the  whole- 
sale dealer,  and  the  retail  dealer.  Put  the  number  of  units 
of  money  which  represent  the  retail  price  equal  to  $  15.60 
and  find  the  value  of  100  units  of  money,  which  is  the  cost 
of  manufacturing. 


218  ARITHMETIC 

QUESTION  IN  PROOF 

(4)  The  manufacturer  of  an  article  makes  a  profit  of  25%, 
the  wholesale  dealer  a  profit  of  20%,  and  the  retail  dealer  a 
profit  of  30%.  What  is  the  retail  price  of  an  article  which 
cost  the  manufacturer  $  8  ? 

PROOF 

The  manufacturer's  gain  =  25 %  of  $8  =  $2. 

The  manufacturer's  selling  price       =  $10. 

The  wholesale  dealer's  gain  =  20  %  of  $  10  =  $  2. 

The  wholesale  dealer's  selling  price  =  $12. 

The  retail  dealer's  gain  =  30 %  of  $  12  =  $ 3.60. 

The  retail  dealer's  selling  price          =  $  15.60. 

is  the  correct  answer  to  the  previous  question. 


MAKING  QUESTIONS 

(5)  Make  a  question  in  which  you  are  given  the  selling 
price  and  the  gain  per  cent,  and  are  required  to  find  the  cost 
price. 

MAKING 

Let  the  cost  of  a  house  =  $6250. 

Let  the  gain  per  cent  on  selling  =  37 1  %. 

The  gain  =  37£  %  of  $6250  =  $2343.75. 

The  selling  price  =  $6250  +  $2343.75  =  $8593.75. 

Problem 

A  house  was  sold  at  37|-%  above  cost.  If  the  selling  price 
was  $8593.75,  find  the  cost  price. 

Other  questions  may  also  be  written  down  from  the  same 
making,  thus  : 

A  house  which  cost  16250  was  sold  at  a  gain  of  37  J%. 
Find  the  selling  price. 

A  house  which  cost  16250  was  sold  for  $8593.75.  Find 
the  gain  per  cent. 


PERCENTAGE  219 

A  house  which  cost  $6250  was  sold  at  a  gain  of  $2343.75. 
Find  the  gain  per  cent. 

In  the  following  exercise  state  in  general  terms  how  to 
solve  each  question.  Prove  some  of  your  answers  correct, 
framing  the  question  in  proof.  Make  questions  similar  to 
problems  in  the  exercise. 

Exercise  145 

1.  A  lot  of  dry  goods  was  sold  at  an  advance  of  18%.     If  the 
gain  was  $  436.50,  what  was  the  cost  ? 

2.  I  made  a  mixture  of  wine  consisting  of  1  gal.  at  50^,  3  at 
90^,  4  at  $  1.20,  and  12  at  40^.    I  sell  the  mixture  at  $  1.60  a  gal. 
Find  my  gain  per  cent. 

3.  A  merchant's  price  is  25%  above  cost.    If  he  allow  a  customer 
a  discount  of  12%  on  his  bill,  what  per  cent  profit  does  he  make? 

4.  If  cloth  when  sold  at  a  loss  of  25%  brings  $5  a  yd.,  what 
would  be  the  gain  or  loss  per  cent  if  sold  at  $  6.40  a  yd.  ? 

5.  Eggs  are  bought  at  11  i  a  doz.,  and  sold  at  the  rate  of  8  for 
25^.     Find  the  rate  of  profit. 

6.  A  merchant  sells  goods  to  a  customer  at  a  profit  of  60%, 
but  the  buyer 'becomes  bankrupt  and  pays  only  70  cents  on  the 
dollar ;  what  per  cent  does  the  merchant  gain  or  lose  on  the  sale  ? 

7.  A  man  bought  a  horse  which  he  sold  again  at  a  loss   of 
10%.    If  he  had  received  $  45  more  for  him  he  would  have  gained 
121%.     Find  the  cost  of  the  horse. 

8.  A  bookseller  sold  a  book  at  17%  below  cost,  but  had  he 
charged  50  cents  more  for  it,  he  would  have  gained  7%.     Find 
the  cost  of  the  book  to  the  bookseller,  and  the  price  at  which  he 
sold  it. 

9.  A  tradesman  bought  goods  for  $  1200  and  sold  \  of  them  at 
a  loss  of  10%.     For  how  much  must  he  sell  the  remainder  to  gain 
20%  on  the  whole? 


220  ARITHMETIC 

10.  A  man  bought  a  house  and  lot  for  $  4750.     After  spending 
$  1143  on  repairs  and  improvements,  and  paying  $  128  for  taxes 
and  other  expenses,  he  sold  the  property  for  $  6800.     What  rate 
per  cent  of  profit  did  his  investment  yield  him  ? 

11.  By  selling  cloth  at  $1.20  per  yd.,  a  tradesman  lost  6J-% 
on  his  outlay.     At  what  price  must  he  sell  it  to  gain  121  %? 

12.  If  a  manufacturer  sells  an  article  of  which  the  first  cost  is 
$400,  to  a  wholesale  dealer  at  10%  profit,  the  wholesale  dealer 
to  the  retailer  at  15%  profit,  and  the  retailer  to  the  consumer  at 
30%  profit,  what  sum  is  paid  by  the  consumer  as  profits  in  addi- 
tion to  the  first  cost  of  the  article  ? 

13.  A  grocer  sold,  at  51^  per  lb.,  a  portion  of  a  stock  of  tea, 
incurring  a  loss  of  15%  and  a  total  loss  of  $18  on  the  quantity 
sold.     How  many  lb.  did  he  sell  ? 

14.  A  merchant  marks  his  goods  so  that  he  may  allow  a  dis- 
count of  5%,  and  still  make  a  profit  of  15%.     Find  the  marked 
price  of  broadcloth  that  cost  him  $  3.80  a  yd. 

15.  A  drover  bought  400  sheep  at  a  certain  price  per  head. 
He  sold  |  of  them  at  a  gain  of  20%,  T%  of  them  at  a  gain  of  15%, 
and  the  remainder  at  a  loss  of  10%,  gaining  on  the  whole  $217. 
How  much  did  he  pay  for  the  400  sheep  ? 

16.  A  grain  dealer  expended  a  certain  sum  of  money  in  the 
purchase  of  wheat,  half  as  much  again  in  the  purchase  of  barley, 
and  twice  as  much  in  the  purchase  of  oats ;  he  sold  the  wheat  at 
a  profit  of  5%,  the  barley  at  a  profit  of  8%,  and  the  oats  at  a 
profit  of  10%,  receiving  all  together  $9740.     Find  the  sum  laid 
out  in  each  grain. 

17.  A  person  sold  two  horses  at  $160  each,  losing  20%  on  one 
and  gaining  20%  on  the  other.     Did  he  gain  or  lose  on  the  whole 
transaction,  and  how  much  ? 

18.  A   man  bought  360  bu.   of  wheat  at  a  certain  price  per 
bu.  and  sold  J  of  it  at  a  gain  of  10%,  1  at  a  loss  of  25%,  and 


PERCENTAGE  221 

the  remainder  at  a  gain  of  45%,  and  by  so  doing  realized  $594 
for  the  whole  lot.     What  was  the  cost  price  per  bu.  ? 

19.  A  speculator  paid  $  1400  for  two  lots,  the  price  of  one  of 
them  being  40%  that  of  the  other.     He  sold  the  cheaper  lot  at  a 
gain  of  50%,  and  the  dearer  one  at  a  loss  of  30%.     Find  his  gain 
or  loss  per  cent  on  the  whole  transaction. 

20.  A  merchant  buys  3150  yd.  of  cloth.     He  sells  J  of  it  at  a 
gain  of  6%,  ^  at  a  gain  of  8%,  ^  at  a  gain  of  12%,  and  the 
remainder  at  a  loss  of  3%.     Had  he  sold  the  whole  at  a  gain  of 
5%  he  would  have  received  $28.98  more  than  he  did.     Find  the 
prime  cost  of  1  yd. 

21.  The  manufacturer  of  an  article  charged   20%   profit,  the 
wholesale  dealer  charged  25%  of  an  advance  on  the  manufactu- 
rer's price,  and  the  retail  dealer  charged  30%  of  an  advance  on 
the  wholesale  price.     Find  the  cost  to  the  manufacturer  of  an 
article  for  which  the  retail  dealer  charged  $  23.40. 

22.  I  buy  two  cows  for  $  55 ;  if  I  sell  the  first  at  a  loss  of  5%, 
and  the  second  at  a  gain  of  5%,  I  should  gain  T5T%.     What  was 
the  price  of  each  cow  ? 


COMMERCIAL  OR  TRADE  DISCOUNT 

216.  Commercial  discount  is  an  allowance  made  by  mer- 
chants upon  their  catalogue  prices. 

The  commercial  discount  is  reckoned  at  a  certain  rate  per 
cent. 

Sometimes  several  discounts  are  allowed  to  a  purchaser. 

In  such  a  case,  the  first  discount  is  to  be  deducted,  and 
then  the  second  discount  is  to  be  reckoned  upon  the  re- 
mainder and  then  deducted,  and  so  on  for  each  successive 
discount. 


222  ARITHMETIC 

217.  What  is  the  net  amount  of  a  bill  for  1720  subject  to 
discounts  of  20%  and  6%?  Find  a  single  discount  equiva- 
lent to  these  successive  discounts. 

The  first  discount  =  20  %  of  $  720  =  $  144. 
The  first  remainder  =  .$  720  —  $  144  =  $  576. 
The  second  discount  =  6  %  of  $  576  =  $  34.56. 

/.  the  net  amount  =  $  576  —  $  34.56  =  $  541.44. 
Again,  the  single  equivalent  discount  =  $  720  —  $  541.44  =  $  178.56. 
.-.  the  rate  of  a  single  discount  =  $  178.56  -4-  $  720  =  .248  =  24.8  %. 

Exercise  146 

1.  An  invoice  was  $650,  trade  discounts  20%  and  8%  off. 
Find  the  cost  of  the  goods. 

2.  What  is  the  net  amount  of  a  bill  of  goods,  the  list  price  of 
which  is  $245,  trade  discounts  18%  and  5%  off  for  cash?. 

3.  What  is  the  difference  on  an  invoice  of  $  540,  between 
40%  direct  discount,  and  discounts  of  25%  and  15%? 

4.  A  dealer  buys  a  book,  list  price  80^,  at  a  discount  of  25% ; 
he  sells  the  book  for  80^.     What  per  cent  is  the  profit  ? 

5.  What  is  the  net  amount  of  a  bill  of  $  480,  discounts  being 
12|%  and  8%?     Find  a  single  discount  equivalent  to  these  suc- 
cessive discounts. 

6.  A  man  paid  $380  for  goods,  at  discounts  of  20%  and  5%. 
Find  the  list  price  of  the  goods. 

7.  A  dealer  paid  $299.20  for  goods  at  15%  and  12%  off. 
Find  the  list  price  of  the  goods. 

8.  Find  the  net  cash  amount  of  a  bill  for  $  1266,  subject  to 
discounts  of  331%,  10%,  and  5%,  for  cash. 

9.  Find  the  difference  between  a  single  discount  of  40%,  and 
successive  discounts  of  25%  and  15%. 

10.  Find  the  net  amount  of  a  bill  of  $  340,  discounts  being  30, 
15,  and  6.  Find  a  single  discount  equivalent  to  these  three  dis- 
counts. 


PERCENTAGE  223 

11.  Find  the  net  cash  amount  of  a  bill  of  $  254,  discounts  being 
25%,  12^%,  5%.     Find  a  single  discount  equivalent  to  these 
three  successive  discounts. 

12.  A  merchant  who  receives  successive  discounts  of  20%,  15%, 
and  10%,  on  a  bill  of  $  750,  sells  at  an  advance  of  33J%.     What 
does  he  sell  his  goods  for  ?     His  selling  price  is  what  per  cent 
less  than  the  list  price  ? 

13.  What  is  the  difference  between  discounting  a  bill  of  $  3000 
at  40%,  and  then  taking  a  discount  off  the  remainder  of  5%  for 
cash,  and  discounting  the  whole  at  45%? 

14.  A  merchant  buys  goods  at  40  and  20  off  the  list  price  and 
sells  them  at  30  and  10  off  the  list  price.     What  is  his  gain  per 
cent  ? 

15.  An  invoice  of  crockery,  amounting  to  $  1473.20,  was  sold 
Jan.  3,  at  90  days,  subject  to  40%  and  10%  discount,  with  an 
additional  discount  of  6%  if  paid  within  20  days.     How  much 
will  be  required  to  pay  the  bill  on  Jan.  21  ? 

COMMISSION  AND  BROKERAGE 

218.  A  Commission  Merchant  is  one  who  buys  or  sells  goods 
for  other  persons  by  their  authority.  Commission  merchants 
are  usually  placed  in  possession  of  the  goods  bought. 

A  Broker  is  a  person  who,  in  the  name  of  his  principal, 
effects  contracts  to  buy  or  sell. 

The  broker  is  not  in  general  placed  in  possession  of  the 
goods  bought  or  sold. 

The  title  Broker  is  also  applied  to  persons  who  deal  in 
stocks,  bonds,  bills  of  exchange,  promissory  notes,  etc.,  and 
to  mercantile  agents,  who  transact  the  business  for  a  ship  in 
port. 

Commission  is  the  charge  made  by  an  agent  for  transacting 
business. 


224  ARITHMETIC 

In  buying,  the  commission  is  reckoned  on  the  cost  price  , 
in  selling,  the  commission  is  reckoned  on  the  selling  price. 

219.  (1)  A  commission  merchant  sold  270  bbl.  of  flour  at 
$  6  a  bbl.,  and  received  5%  commission.  What  was  his  com- 
mission ?  How  much  did  he  remit  to  his  employer? 

The  selling  price  =  270  x  $6  =  $  1620. 
.-.  the  commission  =  5  %  of  $  1620  =  $  81. 
.-.  the  amount  remitted  -  $  1620  -  $81  =  $  1539. 

(2)  A   commission   of   $  242.58   was    charged   for   selling 
$3772  worth  of  goods.     What  was  the  rate  of  commission? 

o/J.9  CQ 

The  commission  =  — — —  of  the  selling  price 

3772 

=  .0643  of  the  selling  price. 
.-.  the  rate  of  commission  =  6.43%. 

(3)  A  grain  dealer  charged  3|%  for  selling  a  quantity  of 
wheat,  and  received  for  his  commission  f  218.40.     For  how 
much  did  he  sell  the  wheat? 

The  commission  =  3|%  or  .035  of  the  selling  price. 

.035  of  the  selling  price  =  $218.40. 
/.  the  selling  price  =  218.40  -4-  .035  =  $6240. 

(4)  If  $512.50  include  the  price  paid  for  certain  goods, 
and   2|%  commission  to  the  agent,  how  much  money  does 
the  agent  expend  in  purchasing  the  goods  ? 

Let  the  cost  price  of  the  goods  =  100  units  of  money. 

Then  the  commission  =  2|  units  of  money. 

The  amount  sent  to  the  commission  merchant  =  102  \  units  of  money. 

1021  units  =  $512.50. 

1  unit  =  512.50  ~  102.5  =  $5. 
100  units  =  $  500. 
.-.  the  cost  of  the  goods  =  $  500. 

As  in  Exercise  145,  give  the  general  statements  of  solu- 
tions, prove  answers,  and  make  questions.  Do  this  also  in 
each  of  the  following  exercises. 


PERCENTAGE  225 

Exercise  147 

1.  A  commission  merchant  sold  480  bbl.  of  flour  at  $3.50  a 
bbl.  on  a  commission  of  2%.     What  was  his  commission  ?     How 
much  did  he  remit  to  his  employer  ?     The  amount  remitted  was 
what  per  cent  of  the  selling  price  ? 

2.  My  agent  sold  coffee  to  the  amount  of  $  850  on  a  commis- 
sion of  3%.     Find  his  commission  and  also  the  amount  remitted 
to  his  employer.     Tire  amount  remitted  is  what  per  cent  of  the 
selling  price  ? 

3.  An  agent  sold  210  bu.  of  oats  at  60^  a  bu.,  and  charged 
$3.78  for  doing  so.     Find  his  rate  of  commission. 

4.  On  a  debt  of  $1725  a  creditor  receives  a  dividend  of  60%, 
on  which  he   allows   his   attorney  5%.      He   receives   a  further 
dividend  of  25%,  on  which  he  allows  his  attorney  6%.     What  is 
the  net  amount  that  he  receives  ? 

5.  If  a  commission   of  $  212.94  is   paid  for  buying   $  6552 
worth  of  goods,  find  the  rate  per  cent  of  commission. 

6.  An   agent   received   $40.62|-   for   selling  a   house  worth 
$  1625.     Find  his  rate  per  cent  of  commission. 

7.  An  agent,  who  is  paid  a  commission  on  what  he  invests, 
received  $  4896,  and  invests  $  4800.      Find  his  rate  per  cent  of 
commission. 

8.  An  agent  received  $  56  for  selling  grain  on  a  commission 
of  4%.     Find  value  of  grain  sold. 

9.  A  commission  merchant  charged  2|%  for  selling  a  quantity 
of  pork,   arid    received    for  his    commission    $64.82.      Find  the 
selling  price  of  the  pork. 

10.  The  owner  of  a  house  offered  an  agent  $  500  commission, 
if  the  agent  could  sell  the  house  for  $  10,500.  What  rate  per 
cent  commission  was  the  owner  offering  ?  Had  the  owner  offered 
5%  commission,  what  would  have  been  the  commission  ou 
$  10,500  ? 

Q 


226  ARITHMETIC 

11.  I  bought  a  bicycle  for  $  70,  which  was  jr  of  my  commission 
at  3  5%  for  selling  a  quantity  of  land.     For  how  much  was  the 
land  sold  ? 

12.  A  real  estate  dealer  sold  land  for  100  units  of  money,  on 
a  commission  of  4%.     How  many  units  of  money  did  he  keep 
for  his  commission  ?     How  many  units  of  money  did  he  send  his 
employer  ?     If  his  employer  received  $  2880,  what  did  the  land 
sell  for  ?     What  was  the  agent's  commission  ? 

13.  An  agent  remits  $4850  to  his  employer  after  taking  out 
his  commission  of  3%.     Find  the  selling  price. 

14.  My  agent  sent  me  as  my  share  of  the  selling  price  of  flour 
$2038.40.     If   the  flour  sold  for  $3.25  a  bbl.,  and  the  agent's 
commission  was  2%,  how  many  bbl.  did  he  sell  ? 

15.  My  agent  bought  a  quantity  of  goods  for  me  on  a  com- 
mission of  2  % .     If  the  cost  of  the  goods  was  100  units  of  money, 
how  many  units  of  money  did  his  commission  equal  ?    How  many 
units  did  I  have  to  send  him  to  cover  the  cost  of  the  goods  and 
his  commission  ? 

16.  A  merchant  in  Buffalo  sends  a  commission  merchant  in 
New  York  $  3120,  instructing  him  to  purchase  goods,  reserving 
his  commission  at  4%.     Find  his  commission. 

17.  A  merchant  sent  $  3238.30  to  New  Orleans  to  be  expended 
in  cotton.     The  broker  in  New  Orleans  charged  6%  commission. 
What  sum  was  paid  for  the  cotton  ? 

18.  Sent  to  a  commission  merchant  in  Chicago  $  2080.80  to 
invest  in  flour,  his  commission  being  2%  on  the  amount  expended. 
How  many  bbl.  of  flour  could  be  purchased  at  $  4.25  a  bbl.  ? 

19.  A  real  estate  broker  sold  a  house  on  3^%  commission,  and 
sent  to  the  owner  $  6176.     What  was  the  broker's  commission, 
and  what  sum  did  he  receive  for  the  house  ? 

20.  I  send  $  5250  to  a  commission  merchant  in  St.  Louis,  who 
charges  5%  for  investing,  with  instructions  to  purchase  certain 


PERCENTAGE  227 

• 

goods,  deducting  his  commission  from  the  amount  of  money  sent 
him.     Find  his  commission. 

21.  (a)  Received  $4100  from  my  agent,  who  had  deducted  his 
commission  at  5%  as  proceeds  of  sale  of  goods.     What  were  the 
goods  sold  at  ?     (6)  Remitted  $  4100,  including  commission,  to 
my  agent  to  invest  for  me  on  commission  of  5  % .     What  was  his 
commission  ? 

22.  An  agent  sold  a  quantity  of  flour  for  $  5100  on  commission 
of  3%,  and  invested  the  remainder  in  tea,  first  taking  out  his 
commission  of  2%  on  the  price  paid  for  the  tea.     Find  the  amount 
paid  for  the  tea,  and  also  the  total  commission. 

INSURANCE 

220.  Insurance   is  a  contract   by  which  a   person  whose 
property  is  insured  receives  security  against  loss  by  fire  or 
accident  in  consideration  of  a  sum   of  money  paid   to  the 
insurance  company. 

The  Premium  is  the  sum  paid  for  insurance.  It  is  always 
a  certain  per  cent  of  the  sum  insured. 

The  Policy  is  the  written  contract  of  insurance. 

221.  (1)  A  factory  valued  at  $  35,000  was  insured  for  -|  of 
its  value,  the  rate  of  insurance  being  |%  for  one  year.     What 
was  the  amount  of  the  premium  ? 

The  premium  =  f  %  of  f  of  $35,000. 

=  J_x§xi35000  =  $  131.25. 

800      5  1 

(2)  The  sum  of  $  285  was  paid  for  the  insurance  at  f  of 
its  value  of  a  ship  worth  $  50,000.  What  was  the  rate  per 
cent  of  premium  if  $  3.75  was  charged  for  the  policy  and  the 
preliminary  survey  ? 


228  ARITHMETIC 

• 

The  premium  =  .$285  -  $3.75  =  $281.25. 
The  amount  insured  =  f  of  $50,000  =  $37,500. 
The  rate  of  insurance  =  $281.25  +  $37,500  =  .0075. 
.-.  the  rate  per  cent  =  T7j\-,  %  or  f  %. 

(3)  For  what  sum  must  a  cargo  worth  $33,950  be  insured 
at  3%  so  that  in  case  of  loss  the  owner  may  recover  both  the 
value  of  the  cargo  and  the  premium  ? 

Let  the  amount  of  insurance  =  100  units  of  money. 
Then  the  premium  =      3  units  of  money. 
The  value  of  the  goods  =    07  units  of  money. 
97  units  of  money  =  $  33,950. 

1  unit   of  money  =  $  350. 
100  units  of  money  =  -$35,000. 
.-.  the  cargo  must  be  insured  for  $  35,000. 

Exercise  148. 

1.  A  warehouse  valued  at  $  62,500  was  insured  for  f  of  its 
value.     The  rate  of  insurance  was  11%  for  three  years,  and  the 
cost  of  the  policy  and  the  agent's  expenses  were  $  2.50.     What 
was  the  amount  paid  for  the  insurance  ? 

2.  An  insurance  company  took  a  risk  at  21%,  and  reinsured  f 
of  the  risk  at  2%.     The  premium  received  exceeded  the  premium 
paid  by  $  42.     Find  the  amount  of  the  risk. 

3.  What  will  be  the  cost  of  insuring  a  cargo  of  24,000  bu.  of 
wheat  valued  at  $  1.05  per  bu.,  the  insurance  covering  4  of  the 
value  of  the  cargo,  the  premium  rate  being  l-g-%,  and  the  other 
expenses  of  the  insurance  being  2^%  of  the  premium  ? 

4.  A  merchant's   stock   was   insured  for    $  42,000,   ±  Of   this 
amount  being  at  J%,  f  of  the  remainder  at  f%,  and  the  re- 
mainder at  £%.     Find  the  total  amount  of  premium  paid. 

5.  A  merchant  insured  his  stock  for  $  33,000  for  one  year  at 
-J%.     Six  months  thereafter  the  policy  was  cancelled  at  the  re- 
quest of  the  insured.     Find  the  amount  of  premium  returned,  the 
short  rate  for  six  months  being  |%. 


PERCENTAGE  229 

6.  A   factory    and    the    machinery    therein    is    insured    for 
$65,000;  f  of  this  sum  is  at  f  %  premium,  and  the  remainder  is 
at  1%.     What  is  the  average  rate  per  cent  of  premium  paid  on 
the  whole  ? 

7.  A  fire  insurance  company  charged  $  196.88  for  insuring  a 
house  for  f  17,500.     What  was  the  rate  per  cent  of  insurance  ? 

8.  A  merchant's  stock  was  worth  $  120,000.     He  insured  it  at 
|  its  value,  paying  $  700  premium.     What  was  the  rate  per  cent 
of  insurance  ?     What  was  the  rate  in  cents  per  $  100  ? 

9.  A  shipment  of  goods  is  insured  for  $  7500,  and  $  18.75  is 
paid  as  premium.     At  that  rate,  what  would  be  the  amount  of  the 
premium  on  $  18,750  ? 

10.  For  what  sum  was  a  house  insured  if  the  premium  paid 
was  $  17.50  and  the  rate  of  insurance  I  %  ? 

11.  For  what  sum  was  a  shop  insured  if  the  rate  of  insurance 
was  65^  per  $  100  and  the  premium  paid  was  $  81.25  ? 

12.  A  fire  insurance  company  received    $  350  for  insuring  a 
factory  at  1^%  premium,  and  charged  I  %  for  insuring  a  less  haz- 
ardous property  of  the  same  valuation  as  the  factory.     What  was 
the  amount  of  the  premium  paid  on  the  second  property  ? 

13.  A  building  and  contents  are  insured  as  follows:  $12,000 
in  the  first,  $  8000  in  the  second,  and  $  5000  in  the  third  insur- 
ance company.     Were  a  loss   to  the  extent  of   $  3500  to  occur 
through  fire,  what  portion  of  the  loss  should  each  company  bear  ? 

14.  Merchandise  valued  at  $  63,000  was   insured  in  the  first 
insurance  company  for  $  15,000,  in  the  second  for  $  12,000,  and 
in  the  third  for  $  8000.     If  the  merchandise  is  damaged  by  fire  to 
the  extent  of    $  10,500,  how  much  of   the  damage  should  each 
company  pay  ? 

15.  A  fire  insurance  company  insured  a  building  for  $  60,000 
at  I  %  premium,  and  reinsured  \  of  the  risk  in  another  company 
at    s%,   and  ^  of  the  risk  in  a  third  company  at  £%.     What 


230  ARITHMETIC 

amount  and  what  rate  of  premium  did  the  company  net  on  the 
remainder  of  their  risk  ? 

16.  A  steamboat  worth  $  60,000  is  insured  in  three  companies; 
in  two  to  the  amount  of  $  15,000  each,  and  in  the  third  to  the 
amount  of    $  20,000.     For  what    sum  would    each    company  be 
liable  if    the  vessel  were   to   sustain  damage  to  the  extent  of 
$  6600  ? 

17.  For  what  amount  must  property  worth  $  7600  be  insured, 
at  5%,  so  that  in  case  of  loss  both  the  premium  and  the  value  of 
the  goods  may  be  recovered  ? 

TAXES  AND  DUTIES 

222.  A  Tax  is  a  sum  of  money  assessed  on  persons  or  prop- 
erty for  public  purposes. 

The  tax  on  property  is  reckoned  at  a  certain  rate  per  cent 
of  the  assessed  value  of  the  property. 

Direct  taxes  are  levied  by  the  state,  county,  township,  city, 
or  the  school  district. 

Some  states  levy  a  tax  upon  each  voter,  independent  of 
the  property  he  owns.  Such  a  tax  is  called  a  Poll-tax,  and 
as  a  rule  does  not  exceed  $2  a  year. 

Indirect  taxes,  called  Duties,  are  levied  by  the  general  gov- 
ernment on  imported  goods. 

An  Ad  Valorem  Duty  is  reckoned  at  a  certain  rate  per  cent 
of  the  cost  of  the  goods  in  the  country  from  which  they  have 
been  imported. 

A  Specific  Duty  is  a  fixed  charge  on  the  quantity  of  goods 
without  reference  to  their  cost,  as  a  specific  tax  of  one  cent 
a  pound. 

223.  The  people  of  a  school  section  wish  to  build  a  new 
school-house,  which  will  cost  $2850.     The  taxable  property 


PERCENTAGE  231 

of  the  section  is  valued  at  $190,000;  what  will  be  the  rate 
of  taxation,  and  what  will  be  a  man's  tax  whose  property  is 
valued  at -17500? 

The  tax  on  $  190,000  =  $2850. 

.  •.  the  rate  of  taxation  =  $2850  --  $  190,000  =  .015  or  1£%. 
.  •.  the  tax  on  $  7500  =  1£  %  of  $  7500  =  $  112.50. 

Exercise  149 

1.  State  expenses  which  the  government  meets  by  taxation; 
the  county ;    the  township ;    the  village  or  city ;  the  school  dis- 
trict. 

2.  What  is  the  tax  on  property  assessed  at  $  6400,  the  rate  of 
taxation  being  l-J-%? 

3.  In  a  school  section  a  tax  of  $  4000  is  to  be  raised.     If  the 
assessed  valuation  of  the  property  is  $  250,000,  what  will  be  the 
tax  on  the  dollar,  and  what  is  A's  tax,  whose  property  is  valued 
at  $  1800  ? 

4.  What  is  the  assessed  value  of  property  taxed  $  37.50,  at 
the  rate  of  15  mills  on  the  dollar  ? 

5.  What  is  the  assessed  value  of  property  taxed  $  37.80,  at 
the  rate  of  41  mills  on  the  dollar  ? 

6.  A  person,  after  paying  an  income  tax  of  16  mills  on  the 
dollar,  has  $  8265.60  left.     What  is  his  income  ? 

7.  What  amount  must  a  town  be  taxed  so  that  after  allowing 
the  collector  3%  the  net  amount  realized  may  be  $  24,250  ? 

8.  What  sum  must  be  assessed  to  raise  $  12,250,  the  collec- 
tor's commission  being  2%? 

9.  If  it  costs  2%  to  collect,  and  5%  of  the  tax  assessed  is 
non-collectible,  what   amount   must   be  levied  in  order  to  raise 
$  27,930  ? 

10.  The  municipal  rates  being  reduced  from  19 1  mills  to  171 
mills  on  the  dollar,  my  taxes  are  lowered  by  $4.05.  For  how 
much  am  I  assessed  ? 


232  ARITHMETIC 

11.  In  a  certain  section  a  school-house  is  to  be  built  at  an  ex- 
pense of  $  8400,  to  be  defrayed  by  a  tax  upon  property  valued 
at  $  700,000.     What  is  the  rate  of  taxation  to  cover  both  the  cost 
of  the  school-house  and  the  collector's  commission  at  4%? 

12.  The  assessed  valuation  of   a  town  is  $  972,250,  and  the 
town  has  320  polls  paying  $1.50  each;  what  is  the  rate  of  taxa- 
tion when  the  tax  levy  is  $  19,925  ?     What  tax  must  a  person 
pay  whose  property  is  assessed  for  $  7500,  and  who  pays  for  one 
poll? 

13.  If   the  assessed  value  of   a  town  is  $  1,260,000,  and  the 
town  has  420  polls. paying  $  1.25  each,  what  is  the  rate  of  taxa- 
tion on  property  when  the  tax  levy  is  $  29,925  ?     What  does  A 
pay,  whose  property  is  assessed  at  $  8500  and  who  pays  one  poll  ? 

14.  A  house  assessed  at  f  2200  was  rented  for  $  23  a  month, 
the   tenant   to  pay  taxes  and  water-rates.     The  taxes  were  17| 
mills  on  the  dollar,  and  the  water-rates  were  $  5  per  quarter  year. 
How  much  all  together  did  the  tenant  pay  per  year  for  the  house  ? 
If  the  property  had  cost  the  landlord  $  2500,  what  rate  per  cent 
per  year  was  he  receiving  on  his  investment  ? 

15.  Paid  30%  duty  on  a  watch,  and  sold  it  at  a  loss  of  5%  ; 
but  had  it  been  sold  for  $  21.06  more,  there  would  have  been  a 
gain  of  8-|-%.     Find  the  cost  price. 

16.  Distinguish  specific  and  ad  valorem  duties.     A  quantity  of 
raisins  invoiced  at  $  877  cost  $  990.25  in  store,  after  paying  duty 
and  $  16.12  for  freight.     Find  rate  of  duty. 

17.  An   importer   purchased    goods,   paying  freight  10%  and 
duty  20%   on  the   original    outlay;    he  was  obliged  to  sell  the 
goods  at  a  loss  of  20%,  but  had  he  received  $585  more  than  he 
actually  sold  them  for,  he  would  have  made  a  profit  of  4%.    Find 
the  original  cost  of  the  goods. 

18.  What  is  the  duty  on  600  yd.  of  cloth  invoiced  at  6  francs 
per  yd.,  the  duty  being  30%?     (1  franc  =  19.3^.) 


PERCENTAGE  233 

Miscellaneous  Exercise  150 

1.  If  I  buy  an  article  for  $3.60  and  sell  it  for  $4.20,  what  is 
my  gain  per  cent  ? 

2.  If  I  sell  goods  for  $3360  and  gain  12%,  what  was  the  cost 
price  ? 

3.  If  425  yd.  of  silk  be  sold  for  $1657.50,  and  20%  profit  be 
made,  what  did  it  cost  per  yd.  ? 

4.  If,  by   selling   goods   for  $1088,   I  lose  16%,  how  much 
per  cent  should  I  have  lost  or  gained  if  I  had  sold  them  for 
$  1344  ? 

5.  A  tradesman's  prices  are  25%   above   cost  price.     If  he 
allow  a  customer  8%  on  his  bill,  what  profit  does  he  make  ? 

6.  If  8%  be  gained  by  selling  a  piece  of  ground  for  $8251.20, 
what  would  be  gained  per  cent  by  selling  it  for  $  8404? 

7.  Find  the  brokerage  on  $  1324  at  \%. 

8.  Find  the  brokerage  on  $  375  at  5%. 

9.  What  amount  of  money  was  invested,  when  the  broker's 
charges  at  1^%  amounted  to  $  150  ? 

10.  My  agent  has  purchased  real  estate,  on  my  account,  to  the 
amount  of  $  19,384.     What  is  his  commission  at  1^%? 

11.  I  send  my  agent  $3654,  with  instructions  to  deduct  his 
commission  at  1|%,  and  invest  the  balance  in  tea.     How  much 
did  he  invest  ? 

12.  Gave  $  15,037.50  to  a  broker  to  invest,  with  instructions, 
after  deducting  brokerage  at  J%,  to  invest  the  balance  in  govern- 
ment bonds.     What  will  be  the  sum  invested,  and  how  much  will 
be  the  brokerage  ? 

13.  What  will  be  the  premium  of  insurance  on  the  furniture 
of  a  house  valued  at  $1500,  at  2i%? 

14.  What   is   the   premium   for   insuring   a   cargo,  valued  at 
$16,450,  at  3i%  ? 


234  ARITHMETIC 

15.  A  person  at  the  age  of  40  insures  his  life  in  each  of  two 
offices  for  $4500,  the  premiums  being  at  the  rate  of  3J  and  3J% 
respectively.     Find  his  annual  payment. 

16.  A  trader  gets  600  bbl.  of  flour  insured  for  80%  of  its  cost, 
at  21%,  paying  $37.80  premium.     At  what  price  per  bbl.  did  he 
purchase  the  flour  ? 

17.  A  shipment  of  dry  goods  was  insured  at  If  %  to  cover  J  of 
its  value.    The  premium  was  $  28.    What  were  the  goods  worth  ? 

18.  A  man  who  owns  $12,750  worth  of  property  pays  a  tax  of 
$  216.75.     Find  the  rate  on  the  dollar. 

19.  A  certain  town  has  property  assessed  at  $520,000,  and 
levies  a  tax  of  $7800.     What  should  B  pay,  whose  property  is 
assessed  at  $  2500  ? 

20.  A  town  has  levied  a  tax  of  $  7690,  which  sum  includes  the 
amount  voted  for  building  a  town  hall  and  the  collector's  fees,  at 
3%.     What  was  expended  on  the  town  hall  ? 

21.  What  is  the  rate  per  cent  of  commission  when  I  receive  $  5 
for  selling  goods  to  the  value  of  $  180  ? 

22.  I  sold  a  quantity  of  goods  for  $  273.68,  on  a  commission  of 
2f  %.     Find  my  commission. 

23.  A  and  B  insure  their  houses  against  fire,  and  A  has  to  pay 
$  7.50  more  than  B,  who  pays  $  28.75.     Find  the  value  of  their 
houses,  the  rate  of  insurance  being  f  %. 

24.  A.  B.  bought  goods  amounting  to  $  7460  subject  to  25  and  5 
off,  $3730  subject  to  30  off,  and  $  1492  subject  to  20  and  10  off. 
Find  the  net  cost  of  the  goods.    Were  the  invoice-clerk  to  bill  A.  B. 
with  goods  amounting  to  $  12,682  subject  to  30  off,  what  would 
be  the  amount  of  the  error  in  the  net  cost  of  the  goods  ? 

25.  A  mixture  of  coffee  and  chiccory  in  the  proportion  of  8 
parts  of  coffee  to  1  part  of  chiccory  is  sold  at  35^  per  lb.,  being  an 
advance  of  40%  on  the  cost.    The  chiccory  cost  9^  per  lb.    Find  the 
cost  of  the  coffee  per  lb. 


PERCENTAGE  235 

26.  A  man  bought  a  house  and  lot  for  $  4750.     After  spending 
$  1143  on  repairs  and  improvements,  and  paying  $  128  for  taxes 
and  other  expenses,  he  sold  the  property  for  $  6800.     What  rate 
per  cent  of  profit  did  his  investment  yield  him  ? 

27.  In  an  examination  A  obtained  78%  of  the  full  number  of 
marks,  beating  B  by  16%  of  the  full  number.     If  A  received  975 
marks,  how  many  did  B  receive  ?     What  percentage  of  B's  num- 
ber was  A's  ? 

28.  By  selling  a  certain  book  for  $3.96,  I  would  lose  12%  of 
the  cost.     What  advance  on  this  proposed  selling  price  would 
give  a  profit  of  12%  of  the  cost?     What  rate  per  cent  on  the 
proposed  selling  price  would  this  advance  be  ? 

29.  Goods  are  sold  at  a  loss  of  15%  on  the  cost.     By  what 
percentage  of  itself  should  the  selling  price  be  advanced  to  yield 
a  profit  of  15  %  on  the  cost  ? 

30.  A  man  having  bought  a  certain  quantity  of  goods  for  $  150, 
sells  ^  of  them  at  a  loss  of  4%.     By  what  increase  per  cent  must 
he  raise  that  selling  price  that  by  selling  the  whole  at  that  in- 
creased rate  he  may  gain  4%  on  his  entire  outlay? 

31.  The  cost  of  freight  and  insurance  011  a  certain  quantity  of 
goods  was  15%,  and  that  of  duty  10%  on  the  original  outlay. 
The  goods  were  sold  at  a  loss  of  5%,  but  had  they  brought  $3 
more  there  would  have  been  a  gain  of  1%.     How  much  did  they 
cost? 

32.  A  man  began  business  with  a  certain  capital ;  he  gained 
20%  the  first  year,  which  he  added  to  his  capital,  and  37|%  the 
second  year,  which  he  added  to  his  capital ;  in  the  third  year  he 
lost  40  %  ;  had  he  received  $  600  more  for  the  goods  sold  the  last 
year,  he  would  have  cleared  in  the  three  years  2  per  cent  of  his 
original  capital.      Find  the  capital  with  which  he  commenced 
business. 

33.  A  merchant  bought  400  Ib.  of  tea  and  1600  Ib.  of  coffee, 
the  cost  of  the  latter  per  Ib.  being  16|%  that  of  the  former; 


236 


ARITHMETIC 


he  sold  the  tea  at  a  profit  of  33J%,  and  the  coffee  at  a  loss  of 
20%,  gaining,  however,  on  the  whole  $  60.  Find  his  buying 
prices  and  his  selling  prices. 

34.  A  sells  a  quantity  of  wheat  at  $  1  per  bu.  and  gains  20%. 
Afterwards  he  sold  some  of  the   same  wheat  to  the  amount  of 
$37.50  and  gained  50%.     How  many  bu.  were  there  in  the  last 
lot  and  at  what  rate  per  bu.  did  he  sell  it  ? 

35.  A  person  marks  his  goods  so  that  he  may  allow  a  discount 
of  4%,  and  still  make  a  profit  of  15%.     What  must  be  the  marked 
price  of  an  article  that  cost  him  $  4.80  ? 

36.  A  manufacturer  who  employed  men  at  $1.60  a  day  found 
that  he  could  save  15%  by  employing  women.     What  wages  were 
paid  the  latter,  supposing  a  man  could  do  -^  more  than  a  woman 
in  the  same  time  ? 

37.  A  merchant  buys  goods;  the   cost  of  freight  is  8%,  and 
that  of  insurance  12%  on  the  original  outlay  ;  he  is  obliged  to  sell 
them  at  a  loss  of  7%  ;  but  if  he  had  received  $  5.10  more  for  them 
he  would  have  gained  1  J%.     Find  the  original  outlay. 

38.  A  merchant  sells  50  yd.  of  broadcloth  at  a  gain  of  15%, 
and  75  yd.,  which  cost  the  same  per  yd.,  at  a  gain  of  10%,  and 
finds  that  if  he  had  sold  the  whole  at  a  uniform  gain  of  12J%,  he 
would  have  received  $  2.25  more  than  he  actually  did  receive. 
WThat  was  the  cost  price  per  yd.  ? 

39.  A   man   buys  goods  for  a  certain   sum,   and  m?aiks  ^  of 
them  at  a  profit  of  24%,  and  f  of  them  at  a  profit  of   36%  ; 
but  had  he  marked   f  of  them  at  24%  gain,  and  -J  at  36%  gain, 
he  would  have  realized  $  240  less  than  before.     Find  the  cost  of 
the  goods. 

40.  A  wheat  buyer  sold  i  of  his  wheat  at  a  certain  gain  per 
cent,  i-  of  it  at  a  gain  of  twice  the  former  rate  per  cent,  and  the 
remainder  at  a  gain  per  cent  of  3  times  the  first  gain.     If  the 
gain  on  the  entire  stock  was  26%,  what  did  he  gain  on  each  part  ? 


PERCENTAGE  237 

If  he  gained  5%  °n  the  first  part,  what  was  the  entire  gain  per 
cent  ? 

41.  A  merchant  wishes  to  mark  some  goods  which  cost  $  1.20 
per  yd.,  so  that  after  making  a  reduction  of  20%  off  the  marked 
prices,  he  may  yet  gain  10%.  At  what  price  per  yd.  must  he 
mark  the  oods  ? 


42.  Sold  goods  to  a  certain  amount  on  a  commission  of  5%,  and 
having  remitted  the  net  proceeds  to  the  owner,  received  for  prompt 
payment  ^-%,  which  amounted  to  $  24.22|.     What  was  the  selling 
price  of  the  goods  ? 

43.  My  agent  sold  a  quantity  of  flour  for  $  2550  on  a  commis- 
sion of  3%,  and  invested  the  proceeds  (after  taking  out  his  com- 
mission) in  tea  on  a  commission  of  2%  on  the  price  paid  for  the 
tea.      Find  how  much  he  paid  for  the  tea  and  also  his  total 
commission. 

44.  At  21%,  for  what  must  property  worth  $  3600  be  insured, 
so  that  in  the  event  of  loss  the  worth  of  the  goods  and  the  premium 
of  insurance  may  be  recovered  ? 

45.  What  sum  must  be  insured  on  a  house  worth  f  665,  so  that 
in  case  of  loss  the  owner  may  receive  ^  of  this  sum,  and  also  |  of 
the  premium,  which  was  at  6%? 

46.  What  single  discount  is  equivalent  to  successive  discounts 
of  20%  and  10%? 

47.  Show  that  successive  discounts  of  specified  rates  may  be 
taken  off  a  list  price  in  any  order  without  affecting  the  net  price. 
Thus  20  and  10  off  is  equivalent  to  10  and  20  off,  so  also  30  and 
10  and  5  off,  10  and  30  and  5  off,  and  5  and  30  and  10  off  are  all 
equivalent. 

48.  An  agent  sold  6  mowing-machines  at  $  120  each,  and  12  at 
$  140  each.     He  paid  for  transportation  $  72,  and,  after  deducting 
his  commission,  remitted  $  2208  to  his  employer.     What  was  the 
rate  of  commission  ? 


238  ARITHMETIC 

49.  A  man  allows  his  agent  5%  of  his  gross  rentals,  and  receives 
a  net  rental  of  $  3488.40.     If  the  gross  rental  is  6%  of  the  value 
of  the  property,  what  is  the  value  of  the  property  ? 

50.  A  shipment  of  goods  is  insured  for  $  6000,  which  sum 
covers  the  value  of  the  goods,  the  premium  at  11%  and  $2.50  for 
expenses.     What  was  the  value  of  the  goods  ? 


CHAPTER   XIV 

INTEKEST 

224.  Interest  is  money  paid  for  the  use  of  money. 
The  Principal  is  the  sum  loaned. 

The  Amount  is  the  sum  of  the  principal  and  interest. 
The  Rate  of  Interest  is  always  expressed  as  a  rate  per  cent 
of  the  principal. 

The  unit  of  time  is  1  yr. 

225.  (1)  What  is  the  interest  on  $  638  for  1  yr.  at  6%  ? 

$  638  principal 

.06  rate  per  unit 
'$38.28  interest  for  1  yr. 

.-.  the  interest  on  $638  for  1  yr.  at  6%  =  6%  of  $638  =  $38.28. 

(2)  Find  the  interest  on  and  the  amount  of  $473.28  for 

81  da.  at  1%. 

$473.28  principal 

.07  rate 

$33.1296  interest  for  1  yr. 

$33.13  interest  for  1  yr. 
9 


40)  $298. 17 

7.45  interest  for  81  da. 
473.28 

$480.73  amount 

The  interest  for  1  yr.  ==  7  %  of  $473.28  ==  $33.13. 

The  interest  for  81  da.  -  ffa  or  &  of  $33.13  ==  $7.45. 

The  amount  -  $  473.28  +  7.45  =  $  480.73. 

239 


240 


ARITHMETIC 


(3)  Find  the  amount  of  1385.35,  from  July  7,  1895,  to 
Oct.  13,  1895,  at  7 


mo. 
Oct.  13  =  10 

July    7=7 


da. 
13 

7 


The  time  =3 


=  3    mo.  = 


yr. 


Thp  ritp  —    4 

-  T5 


ISO/  —  o  o/ 
2    /O  —  z  /o- 

The  interest  =  2  %  of  $  385.35  =  $  6.71. 
The  amount  =  $385.35  +  $6.71  =  $392.06. 

226.    Six  Per  Cent  Method. 

The  interest  at  6%  for  1  yr.    =  .06  of  the  principal. 

The  interest  at  6%  for  1  mo.  =  J.-y  of  .06  or  .005  of  the  principal. 

The  interest  at  C%  for  1  da.    =  JG-  of  .005  or  .000^  of  the  principal. 

Find  the  interest  on  $435  for  9  mo.  24  da.  at  6%. 

The  interest  for  9  mo.  =    9  x  .005    =  .045 

The  interest  for  24  da.  =  24  x  .000  £  =  .004 

The  interest  for  9  mo.  24  da.  =  ^049 

.-.  the  interest  =  .049  x  $435  =  $21.315. 

To  find  the  interest  at  any  other  rate  per  cent,  divide  the 
interest  at  6%  by  6  and  multiply  by  the  given  rate  per  cent. 

The  interest  at  7|%  may  be  found  by  increasing  the  interest  at  6%  by 
li  i  11 

i  or  -  ;  that  at  5.1  %  by  diminishing  the  interest  at  6  %  by  -  or  — 
64  6       12 

By  what  fraction  must  the  interest  at  6%  be  increased  in  order  to  give 
the  interest  at  each  of  the  following  rates  :  7  %,  8  %,  9  %,  6\  %,  6.2  %  ? 

By  what  fraction  must  the  interest  at  6  %  be  diminished  in  order  to  give 
the  interest  at  each  of  the  following  rates  :  5  %,  4  %,  3  %,  4^  %,  4  1  %,  5|  %  ? 

Exercise  151 


Find  the  interest  on : 

1.  $ 449  for  1  yr.  at  5%. 

2.  $757  for  1  yr.  at  4%. 

3.  $643.17  for  1  yr.  at  7%. 

4.  $  725  for  4  mo.  at 


5.  $  587.50  for  5  mo.  at  6% 

6.  $  628.90  for  9  mo.  at  41 

7.  $  323.75  for  60  da.  at 

8.  $  958.50  for  90  da.  at 


INTEREST  241 

9.    $2865  for  33  da.  at  6%.          12.    $225.90  for  63  da.  at  7%. 

10.  $312.80  for  93  da.  at  6%.        13.    $  390.50  for  93  da.  at  6%. 

11.  $  612.94  for  33  da.  at  1\  %.     14.    $  8396.40  for  123  da.  at  8%. 

15.  $  4087.50  f or  1  in o.  3  da.  at  9%. 

16.  $  1465.53  for  3  mo.  3  da.  at  5%. 

17.  $  1350  for  3  mo.  21  da.  at  7%. 

18.  $295.36  for  57  da.  at  6.2%. 

19.  $  1200  from  May  7  to  June  6  at  7%. 

20.  $  975.65  from  Sept.  16  to  Dec.  8  at  6\%. 

21.  $450  from  Sept.  4  to  Oct.  27  at  7%. 

22.  $  79.50  from  Dec.  23  to  Feb.  20  at  74  %. 

23.  $586.67  from  Jan.  15,  1892,  to  May  1,  1892,  at  8%. 

24.  State  how  to  find  the  simple  interest  when  the  principal, 
rate  per  cent,  and  time  are  given. 

25.  Name  the  terms  in  problems  in  Profit  and  Loss,  Commis- 
sion and  Insurance,  which  correspond  to  principal  and  rate  per 
cent  of  interest. 

26.  Find   the   relation   between   the   interest,    principal,    and 
amount,  when  the  time  is  3  mo.,  and  rate  8%  ;    time  120  da., 
rate  9%. 

27.  Find  the  amount  of  $473.28  for  3  mo.  at  \%  per  month. 

28.  Find  the  amount  of  $  885.85  for  1  mo.  15  da.  at  5%. 

29.  Find  the  amount  of  $  628.25  for  185  da.  at  4$  %. 

30.  Find  the  amount  of  $  935.68  for  66  da.  at  6-1  %. 

31.  Find  the  amount  of  $  147.50  for  93  da.  at  7%. 

32.  Find  the  amount  of  $  250  from  July  9  to  Aug.  18  at  S%. 

33.  Find  the  amount  of  $2394  from  May  8  to  Sept.  21  at  4%. 

34.  Find  the  amount  of  $  5246  from  March  1  to  Aug.  3  at  5%. 

35.  Find  the  amount  of  $  230.80  from  Jan.  4,  1896,  to  June  23, 
1896,  at  6%. 

36.  Find  the  amount  of  $657.60  from  Aug.  9  to  Dec,  5  at  8%. 

R 


242 


ARITHMETIC 


37.  A  person  loaned  $  480  for  2  nio.  and  13  da.  at  9%.     What 
interest  did  he  receive  ? 

38.  On  March  20,  a  merchant  sold  goods  to  the  value  of  $  1168, 
and  received  a  note,  due  June  8,  next,  for  that  sum  with  interest 
at  7  %  per  annum.     For  what  amount  was  the  note  drawn  ? 

39.  A  debt  of  $  175  became  due  on  June  13,  after  which  date 
interest  was  charged  at  the  rate  of  8%.     What  must  be  paid  to 
settle  the  debt  Sept.  14  ? 

40.  A  owes  $15,000  bearing  interest  at  5%  per  annum;  he 
pays  at  the  end  of  each  year  for  interest  and  part  payment  of 
principal  $  2500.     Find  the  amount  of  his  debt  at  the  end  of  the 
third  year. 

41.  A  man  engaged  in  business  with  a  capital  of  $10,920  is 
making  12J%  per  annum  on  his  capital,  but  on  account  of  ill 
health  he  quits  the  business  and  loans  his  money  at  7|%.     How 
much  does  he  lose  by  the  change  in  2  yr.  5J  mo.  ? 

42.  $420.  CHICAGO,  June  4,  1895. 

Sixty  days  from  date  I  promise  to  pay  Samuel  Jones,  or  order, 
four  hundred  and  twenty  dollars,  with  interest  at  six  per  cent, 
value  received.  RICHARD  WALSH. 

What  is  the  amount  of  this  note  on  maturity,  63  da.  after 
June  4  ? 

EXACT  INTEREST 

227.  In  order  to  find  the  exact  interest  we  must  reckon 
365  da.  to  a  year.      Exact  interest  is  used   by  the  United 
States  Government  and  sometimes  in  business  transactions. 

228.  The  exact  interest  at  5  %  for  1  da.  is  3 17,  or  7xf  of  the  principal. 
The  common  interest  is  gf ^  or  ^  of  the  principal.     Therefore  the  exact 

interest  is  ^  -H  7\  or  £f  of  the  common  interest.    Hence  the  exact  interest  is 
equal  to  the  common  interest  diminished  by  ^  of  itself. 

229.  Find  the  exact  interest  on  14250   from  May  12  to 
Oct.  3  at  7  % . 


INTEREST  248 

The  number  of  days  from  May  12  to  Oct.  3  =  19  +  30  +  31  +  31  +  30  + 
3  =  144, 

The  interest  on  $  4250  at  7  %  for  1  yr.  =  $  297.50. 

The  interest  on  $4250  at  7  %  for  144  da.  -  |||  of  $297.50  =  $  117.37. 

Exercise  152 

Find  the  exact  interest  on : 

1.  $  2450  for  146  da.  at  6%. 

2.  $  3475  for  292  da.  at  1%. 

3.  $1560  for  60  da.  at  5%. 

4.  $629  for  113  da.  at  6%. 

5.  $  1400  from  July  6  to  Dec.  4  at  5 

6.  $  1850  from  March  1  to  Aug.  6  at 

230.  In   Exercise  151  we   were  given  the  principal,  rate 
per  cent,  and  time,  and  were  required  to  find  the  interest  or 
the  amount.     In  the  following  exercise  we  shall  have  given 
the  principal,  interest  or  amount,  and  the  time,  and  will  be 
required  to  find  the  rate  per  cent. 

231.  (1)  At  what  rate  per  cent  will   1480  yield  $18.20 
interest  in  7  mo.  ? 

The  interest  on  $480  for  7  mo.  -  $  18.20. 

The  interest  on  $  480  for  1  mo.  -r-  $2.60. 

The  interest  on  $480  for  1  yr.  —  $31.20. 

.-.  the  rate  per  cent  =  $31.20  -=-  $480  =  .06$  or  6J%. 

(2)  At  what  rate  per  cent  must  I  loan  $  2840  for  145  da. 
to  amount  to  $2922.36? 

The  interest  =  $  2922.36  -  $  2840  =  $  82.36. 
The  interest  for  145  da.  or  ff  yr.  -  $82.36. 
The  interest  for  1  yr.  =  $82.36  x  If  ==  $204.48. 
.-.  the  rate  per  cent  =  $  204.48  H-  $2840  =  .072  or  7£  %. 


244  ARITHMETIC 

Exercise  153 

1.  A  gentleman  invests  $2500  for  his  son  in   order  to  give 
him  a  yearly  income  of  $  175.     Find  the  rate  per  cent. 

Find  the  rate  per  cent : 

2.  When  the  interest  on  $  1200  for  8  mo.  is  $48. 

3.  When  the  interest  on  $  640  for  9  mo.  is  $  35. 

4.  When  the  interest  on  $  585  for  1  mo.  18  da.  is  $  4.94. 

5.  When  a  loan  of  $600  amounts  to  $626.75  in  7  mo.  4  da., 
what  rate  per  cent  is  charged  ? 

6.  A  man  pays  $14.91  for  the  use  of  $568  for  4  mo.  15  da. 
Find  the  rate  per  cent. 

7.  A  man  lent  $4800  for  6  mo.  6  da.,  and  at  the  expiration 
of  the  time  received  in  payment  of  interest  and  principal  $  4955. 
Find  the  rate  per  cent. 

8.  At  what  rate  will  $438  borrowed  on  April  17  amount  at 
simple  interest  to  $  445.519  on  July  29  next  following,  if  exact 
interest  is  reckoned  ? 

232.    To  find  the  time  when  we  are  given  the  principal, 
interest,  or  amount,  and  rate  per  cent. 

In  what  time  will  the  interest  on  $845  be  $32.951  at  6|  %  ? 

The  interest  on  $  845  at  6J  %  for  1  yr.  =  $  54.92 J. 
Therefore  the  fraction  of  a  year  =  $32.955  -^-  $54.925  =  f. 

/.  the  time  =  f  yr.  or  7  mo.  6  da. 

Exercise  154 

1 .  In  what  time  will  the  interest  'on  $  750  at  7  %  equal  $  26.25  ? 

2.  In  what  time  will  $  400  amount  to  $  415  at  5%  per  annum  ? 

3.  A  man  invests  $4760  at  8%,  in  order  that  his  son  may 
have  an  income  of  $  95.20  at  certain  equal  intervals  of  time.    How 
often  is  the  interest  payable  a  year  ? 


INTEREST  245 

4.  A  gentleman  gives  his  note  for  $  350,  together  with  interest 
at  4|%.     If  he  pays  $362.60  to  settle  the  note,  when  is  it  paid  ? 

5.  A  person  leaves  unpaid  a  sum  of  money,  on  which  he  pays 
7^%  interest,  until  the  interest  equals  ^  of  the  principal.     Find 
the  time. 

6.  A  principal  of  $1200  was  loaned  May  12,  1892,  at  8%. 
At  what  date  did  it  amount  to  $  1216.80  ? 

7.  In  what  time  will  $273.85  yield  $8.86  simple  interest  at 


8.  In  how  many  da,  will  $  733.65  amount  to  $  7-13.70  at  5% 
simple  interest  ? 

9.  A  debt  of  $  175  became  due  on  June  13,  after  which  date 
interest  was  charged  at  7  %  per  annum ;   when  the  debt  was  paid 
the  interest  accrued  on  it  was  $  4.10.     When  was  the  debt  paid  ? 

10.    In  what  time  will  $  143  amount  to  $150  at  7%  interest  ? 

233.    To  find  the   principal  when  the  interest,  time,  and 
rate  are  given. 

Find  the  principal  that  will  produce  $  40.77  in  9  mo.  at 

. 

The  interest  for  9  mo.  or  ^  yr.  i=  $40.77. 

The  interest  for  \  yr.  =  $  13.51). 

The  interest  for  1  yr.  =  $  54.30. 

8%  or  .08  of  the  principal  =  $54.36. 
.-.  the  principal  =  $  54.36  -4-  .08  =  $679.50. 

Exercise  155 

1.    What  principal  will  produce  $60  in  216  da.,  at  7%%  Per 


annum  ? 


2.    A  man  borrowed  money  at  6%,  and  paid  $323.70  interest 
a  yr.     Find  what  sum  he  borrowed. 


246  ARITHMETIC 

3.  A  man    loans   money  at   8%   per  annum,  interest  payable 
semiannually.      If  his  semiannual   interest  is  $  38.40,  find  the 
sum  loaned. 

4.  A  man  left  to  his  wife  a  yearly  income  of  $  1750,  to  the 
oldest  son  $  1540  yearly,  and  to  the  youngest  $  1260  yearly.    Find 
what  sum  must  be  invested  at  Q^%  to  produce  these  amounts. 

5.  What  principal  will  yield  $  43.25  interest  in  J-  yr.  at  5 


234.    To  find  the  principal  when  the  amount,  time,  and 
rate  per  cent  are  given. 

Find  the  principal  that  will  amount  to  $  1312.  50  in  8  mo. 

at  7J%. 

The  interest  on  $  1  for  8  mo.  at  7^%  =  $  .05. 

The  amount  of  $  1  for  8  mo.  at  7i%  =  $  1.05  =  1.05  of  $  1. 

Hence  $  1312.50  is  the  amount  of  $  1312.50  -4-  1.05  =  $  1250. 

/.  the  principal  =  $  1250. 

Exercise  156 

1  .    Find  what    sum   would    pay  now  a  debt  of    $  450  due  in 
6  mo.  at  6%  per  annum. 

2.  What  must  be  paid  now  to  cancel  a  debt  of  $  1368.25  9  mo. 
before  it  is  due,  money  being  worth  7%? 

3.  Which  is  cheaper,  lumber  bought  at  $  35  a  thousand  on 
9  mo.  credit,  or  at  $  34.30  on  6  mo.  credit,  money  being  worth 


4.  I  bought  a  lot,  paying  $  400  cash,  and  the  balance  $  800 
in  9  mo.     What  is  the  cash  value  of  the  lot,  money  bringing 

5.  What  principal  will  amount  to  $  1000  in  4  mo.  at  41  ? 

6.  What  principal  will  amount  to  $  73.56  in  63  da.  at  S%? 

7.  A  debt  due  011  March  3  was  not  paid,  and  interest  at 


was  charged  on  it  from  that  date.     On  June  9  following,  the  debt 
amounted  to  $  100.     What  was  the  sum  due  on  March  3  ? 


INTEREST  247 

8.  A  merchant  bought  500  bbl.  of  flour  at  $  6.25  a  bbl.  on  a 
credit  of  8  mo.     He  sold  it  at  $  6.50  a  bbl.  on  a  credit  of  4  mo. 
What  was  his  net  cash  gain,  money  being  worth  6%? 

9.  A  merchant  borrows  $1600  for  1  yr.  at  7%.     Find  what 
he  owes  at  the  end  of  the  yr.     In  case  he  pays  only  $  12  inter- 
est, how  much  will  he  owe  at  the  beginning  of  the  next  yr.  ? 
\Yhat  will  he  owe  at  the  end  of  the  yr.  ? 

10.  If  I  borrowed  $1200  Jan.  1,  1894,  at  6%,  what  would 
I  owe  Jan.  1,  1895  ?  If  I  kept  the  money  until  Jan.  1,  1896, 
what  would  I  then  owe  ? 

BANK  DISCOUNT 

235.  A  merchant,  who  desires  to  obtain  a  loan  of  8800 
for  90  da.,  makes  a  note  and  takes  it  to  the  bank,  which 
deducts  the  interest  on  $  800  for  93  da.  at  a  certain  rate  per 
cent,  which  varies  from  time  to  time.     This  bank  gives  him 
the  proceeds,  and  collects  the  $800  at  the  end  of  93  da. 

The  3  da.  added  to  the  specified  time  are  called  days  of 
grace,  which  must  elapse  before  payment  is  due. 

California,  Idaho,  New  Jersey,  New  York,  Oregon,  Utah, 
Vermont,  and  Wisconsin  have  abolished  days  of  grace. 

236.  Bank  Discount  is,  therefore,  simple  interest  collected 
in  advance  upon  the  sum  due  on  a  note  at  its  maturity. 

Nearly  all  notes  specify  the  place  of  payment.  In  case  the 
place  of  payment  is  not  specified  in  the  note,  it  is  to  be  paid 
at  the  business  office  of  the  maker  of  the  note. 

237.  8450.75.  CHICAGO,  July  3,  1896. 

Sixty  clays  after  date  I  promise  to  pay  to  the  order  of  James 
Smith,  four  hundred  fifty  and  -Jfo  dollars  at  the  First  National 

Bank.     Value  received. 

HORACE  WARD. 

Discounted  July  3,  at  6%.     Find  proceeds. 


248  ARITHMETIC 

The  day  of  maturity  =  63  da.  after  July  3  =  Sept.  4. 

The  discount  =  the  interest  on  $  450. 75  at  6  %  for  63  da.  =  $  4. 73. 

The  proceeds  =  $  450. 75  -  -s  4. 73  =  $  446. 02. 

238.  The  Day  of  Maturity  is  the  day  on  which  the  note 
becomes  legally  clue. 

The  Proceeds  of  a  Note  is  the  sum  of  money  received  for  it 
when  discounted. 

It  is  found  by  subtracting  the  discount  from  the  value  of 
the  note  at  maturity. 

The  Time  to  run  is  the  time  between  the  day  on  which  the 
note  is  discounted  and  the  day  of  maturity. 

Exercise  157 

1.  $600.  CHICAGO,  July  6,   1896. 
Thirty  days  after  date  I  promise  to  pay  to  George  Boies,  or 

order,  six  hundred  dollars,  value  received. 

EGBERT  BROWN. 
Discounted  at  7  %,  July  6,  1896.     Find  proceeds. 

Face  of  Note  Date  of  Note  Time     Eate  of  Discount 

2.  $312.80;      May  13,  1895  ;  90  da.  ;       6%.      Find  proceeds. 

3.  $225.90;      June  14,  1896  ;     2  mo.  ;     1%.      Find  proceeds. 

4.  $100.00;      Feb.   12,  1896  ;  30  da.  ;       5%.      Find  proceeds. 

5.  State  how  to  find  the  proceeds  of  any  note  discounted  at 
once. 

Face  of  Not.-  Date  of  Note  Time     Rate  of  Discount 

6.  $  1.00  ;          Jan.  7,  1894  ;       57  da.  ;     6%.      Find  proceeds. 

7.  In  question  6  what  face  value  would  give  $  99  proceeds  ? 
$  198  ?     $  495  ?     Prove  your  face  value  correct  by  discounting. 

8.  Write  the  notes  corresponding  to  questions  2  and  3. 

9.  $390/^.  SPRINGFIEL-D,   ILL.,    May  1,   1890. 
Three   months   after  date    I  promise  to  pa}^   to  the  order  of 

Thomas  A.  Stuart,  three  hundred  ninety  and  T5^j  dollars.     Value 

JAMES  HENDERSON. 
Discounted  May  1,  1890,  at  6%.     Find  proceeds. 


INTEREST  249 

239.    (1)    $712.65.  CHICAGO,  July  6,  1895. 

Sixty  days  from  date  I  promise  to  pay  George  Wilson, 
or  order,  seven  hundred  twelve  and  ^f$  dollars,  for  value 

received.  SAMUEL  JONES. 

Discounted  at  7  %,  Aug.  G,  1895. 

In  the  above  note,  find  the  day  of  maturity,  the  time  to  run, 
the  discount,  and  the  proceeds. 

The  day  of  maturity   =  63  da.  after  July  6  =  Sept.  7,  1895. 

The  time  to  run  =  the  number  of  days  between  Aug.  6  and  Sept.  7. 

z32da.  ==¥V_yr. 

The  discount  =  the  interest  on  $  712.65  for  32  da.  at  7  %  =  $  4.43. 

The  proceeds  =  $  712.65  -  $  4.43  =  $  708.22. 

(2)   $450.76.  ST.  Louis,  May  5,  1895. 

Three  months  after  date,  for  value  received,  I  promise  to 
pay  Thomas  King,  or  order,  four  hundred  fifty  and  ^fa  dollars, 
at  the  First  National  Bank,  with  interest  at  Q%. 

ARTHUR  HILL. 

Discounted  July  1,  1895,  at  8%. 

The  day  of  maturity  =  3  mo.  3  da.  after  May  5  =  Aug.  8,  1895. 

The  amount  of  the  note,  Aug.  8,  1895  =  the  amount  of  §450.76  for  3  mo. 

3  da.  at  6%  =  $457. 75. 

The  time  to  run  =  the  number  of  days  between  July  1  and  Aug.  8  =  38  da. 
The  discount       =  the  interest  on  .$  457.75  for  38  da.  at  8  %  =  $  3.87. 
The  proceeds       =  $457.75  -  $  3.87  ==  $453.88. 

In  the  following  exercise  find  the  day  of  maturity,  the  time 
to  run,  the  discount,  and  the  proceeds. 

Exercise  158 

1.    $2400.  CLEVELAND,  0.,  March  3,  1894. 

Three  months  after  date  I  promise  to  pay  Ralph  Barker,  or 
order,  twenty-four  hundred  dollars,  value  received. 

ROBERT  PETERSON. 

Discounted  at  7  %,  May  7. 


250  ARITHMETIC 

2.  A  note  for  $  572.80  drawn  on  June  13,  and  payable  4  mo. 
after  date,  was  discounted  at  7%   on  June  27.     Find  the  pro- 
ceeds. 

3.  $2400.  CLEVELAND,  0.,  March  3,  1894. 
Three  months  after  date  I  promise  to  pay  Ralph  Barker,  or 

order,  twenty-four  hundred  dollars,  for  value  received,  with  inter- 

est at 


ROBERT  PETERSON. 

Discounted  at  7%,  May  7. 

4.  State  business  transactions  which  may  have  preceded  the 
giving  of  the  notes  in  questions  1,  2,  and  3. 

5.  On   July    7,  James    Monroe   bought   a   farm   from   John 
Harris,  paying  $  2000  cash,  and  giving  his  note,  without  interest, 
for  $  1200,  payable  in  60  da.     Write  the  note. 

Face  of  Note  Date  of  Note  Time  Date  of  Disc.  Rate  of  Disc. 

6.  $312.80;  May  13,1890;  90  da.  ;    May  13;  6^%. 

7.  $975.65;  Sept.    5,1892;     3  mo.  ;  Sept.  16  ;  7%. 

8.  $450.00;  Aug.  28,  1891;  60  da.  ;    Sept.  4  ;  7%. 

9.  $79.50;     Dec.    17,1889;     2  mo.  ;  Dec.  23  ;  1\%. 
10.    $  586.67  ;  Dec.   28,  1891  ;     4  mo.  ;  Jan.  15,  1892  ; 


11.  $2480.  BUFFALO,  N.  Y.,  Nov.  19,  1892. 

Six  months  after  date  I  promise  to  pay  Alfred  Jameson,  or 
order,    two   thousand    four   hundred   and   eighty    dollars,    value 

received,  with  interest  at  5%. 

WILLIAM  O'CONNOR. 
Discounted  at  6%,  Jan.  4,  1893. 

12.  $  2065.76.  NEW  ORLEANS,  June  4,  1895. 

Ninety  days  after  date  I  promise  to  pay  to  the  order  of  Edgar 
Johnston,    two   thousand  sixty-five   and   T7^   dollars,   for   value 

received,  with  interest  at  6%. 

ALEXANDER  GRANT. 

Discounted  at  8%,  July  4,  1895. 


INTEREST  251 

13.  State  how  to  find  the  proceeds  of  a  note,  not  bearing  inter- 
est, when  discounted.    What  change  is  to  be  made  in  the  solution 
when  the  note  bea.rs  interest  ? 

14.  Find  the  proceeds  of  a  note  payable  in  87  da.,  whose  face 
value  is  $  1,  discounted  immediately  at  8%. 

15.  In  question  14,  what  would  have  been  the  face  value  of 
the  note  if  the  proceeds  had  been  $  98  ?    $  980  ?   $  196  ?    $  392  ? 

240.    (1)  Find  the  face  of  a  note  payable  in  60  da.,  that 
will  realize  $840  when  discounted  at  6J%. 
Let  us  consider  a  similar  note  whose  face  is  $1. 

The  discount  on  $  1  for  63  da.  at  Q\  %     =  $  .011375. 

The  proceeds  of  a  note  whose  face  is  $  1  =  $  1  --  .011375  =  $  .988625. 

Hence  .988625  of  the  face  =  $  840. 

.-.  the  face  =  $840  -=-  .988625  =  $849.66. 

PROOF 

The  discount  on  $  849.66  for  63  da.  at  6£%  =  $  9.66. 
The  proceeds  =  $  849.66  --  $  9.66  =  $  840. 

.-.  $849.66  is  the  correct  answer. 

(2)  In  solving  the  preceding  question  it  is  unnecessary  to  carry  out  the 
discount  on  $  1  beyond  the  fourth  figure  in  the  decimal. 

Thus  the  discount  on  $  1  =  $  .0114. 
The  proceeds  of  $  1  =  $ .9886. 

The  face  of  the  note         =  $  840  -4-  .9886  =  $  849.68. 

This  is  correct  within  2  $>. 

Exercise  159 

1.  Find  the  face  value  of  a  note  for  27  da.  that  will  realize 
$  1990  when  discounted  at  6%. 

2.  Write  the  note  corresponding  to  question  1,  and  prove  that 
the  face  as  written  in  the  note  is  correct. 

3.  For  how  much  must  a  2  months'  note  be  drawn  so  that 
when  discounted  at  7  %  it  may  yield  $  500  ?     Prove  your  answer 
correct  and  write  the  note, 


252  ARITHMETIC 

4.  A  gentleman  wishes  to  borrow  $  800  at  a  bank.     For  what 
sum  must  he  give  a  60  days'  note  which  is  discounted  at  1\%  ? 

5.  What  must  be  the  face  of  a  4  months'  note  discounted  at 
8%,  to  realize  $89.50? 

6.  For  what  sum  must  a  note  be  drawn  in  order  that  if  dis- 
counted 89  da.  before  maturity,  the  proceeds  may  be  $  425 ;  the 
rate  of  discount  being  7%? 

7.  What  must  be  the  face  of  a  note  so  that  when  discounted 
at  a  bank  for  4  mo.  and  9  da.  at  9%,  it  will  give  $  240  ? 

8.  State  how  to  find  the  face  value  of  a  note,  when  the  term 
of  discount,  the  rate  per  cent,  and  the  proceeds  are  given. 

9.  Make  a  question  in  which  it  is  required  to  find  the  face  of 
a  note  —  given  the  proceeds,  rate  of  discount,  and  term  of  dis- 
count. 

10.  I  owe  a  man  $  575,  and  gave  him  a  note  at  60  da.     Wliat 
must  be  the  face  of  the  note  to  pay  him  the  exact  debt,  when 
discounted  at  (bank  discount)  1^%  a  month? 

11.  A  sold  B  a  bill  of  goods  amounting  to  $  7600,  but  B,  not 
having  the  money,  gave  A  a  note  for  3  mo.,  which,  when  discounted 
at  the  bank  at  8%,  paid  the  debt.     Eequired  the  face  value  of 
the  note. 

Miscellaneous  Exercise  160 

1.  Find  the  interest  on  $  794.35  for  188  da.  at  5%. 

2.  To  what  sum  would   $87.68  amount  in  97   da.  at  6±% 
interest  ? 

3.  A  certain  sum  amounts  to  $  1488  in  8  mo.,  and  $  1530  in 
15  mo.,  simple  interest.     What  is  the  rate  per  cent  ? 

4.  A  merchant  borrowed  $  1680  June  16,  and  $  1728  Sept.  28 ; 
the  merchant  repays  the  whole  sum,  with  interest,  Jan.  2  next. 
Find  the  amount  repaid,  interest  being  74-%  per  annum. 


INTEREST  253 

5.  Bought  9000  bu.  of  wheat  at  75^  a  bu.,  payable  in  6  mo. ; 
I  sold  it  immediately  for  72^  a  bu.,  cash,  and  put  the  money  at 
interest  at  6%.     At  the  end  of  the  6  mo.  I  paid  for  the  wheat. 
Did  I  gain  or  lose  by  the  transaction,  and  how  much  ? 

6.  What  is  the  bank  discount  and  proceeds  of  a  note  of  $  1168, 
drawn  Jan.  18  at  11  mo.,  discounted  at  the  bank  May  20  at  6%? 

7.  I  owe  a  bill  amounting  to  $  219.75,  and  I  give  my  note  for 
60  da.     How  must  I  draw  it  to  cover  the  discount  at  6^%? 

8.  A.  B.  has  a  note  of  $  800  to  pay  at  the  Merchants'  National 
Bank.     At  the  time  of  its  maturity  he  pays  $  200,  and  gives  a 
note  for  3  mo.,  days  of  grace  included,  for  the  balance.     The  rate 
of  discount  being  8  %  per  annum,  what  was  the  face  of  the  note  ? 

9.  A  note  for  $  1750  was  drawn  July  10  for  4  mo.     It  was 
discounted  Sept.  2  at  the  Granite  Bank  at  1\%  per  annum.    What 
sum  was  received  for  it  ? 

10.  A  note  is  drawn  for  3  mo.,  days  of  grace  included,  and 
when   discounted  at  1  %   per  annum  at  the  Farmers'    Bank  it 
realizes  $  4.55  less  than  its  face  value.     What  is  the  face  value 
of  the  note  ? 

11.  Find  the  proceeds  of  the  following  joint  note  discounted 
in  New  York  Dec.  18,  1896,  at  1\%. 

f  347_50_<  NEW  YORK,  Dec.  18,  1896. 

Ninety  days  after  date  we  jointly  and  severally  promise  to  pay 
to  the  order  of  Jno.  Locke  &  Co.,  three  hundred  and  forty-seven 
y5^-  dollars,  at  the  Standard  Bank.  Value  received. 

ISAAC  HARPER. 

A.  C.  EARLY. 

12.  For  how  much  must  a  90-day  note  be  drawn  to  realize 
$  190  when  discounted  at  6%? 

13.  Find  the  proceeds  of   a  note  wliose  face  value  is  $  950, 
payable  in  3  mo.  from  Feb.  1,  1896,  and  discounted  Feb.  6,  1896, 

at  1%. 


254  ARITHMETIC 

14.  A  note  for  $  360  was  discounted  40  da.  before  maturity 
and  the  proceeds  were  $  356.80.     What  was  the  rate  of  discount, 
there  being  no  exchange  ? 

15.  The   proceeds  of   a  note  for  $  137.50,  discounted  40  da. 
before  maturity,  were  $  136.40.     What  was  the  rate  of  discount 
charged  on  the  face  of  the  note  ? 

16.  June  18,  1895,  a  merchant  purchased  goods  amounting  per 
catalogue  prices  to  $  647.80,  subject  to  25  and  5  off.     He  was 
allowed  3  months'  credit,  after  which  he  was  charged  interest 
at  8%.     Find  the  amount  of  the  account  Feb.  21,  1896. 

*  PARTIAL  PAYMENTS 

241.  A  Partial  Payment  is  a  payment  of  only  a  part  of  a 
debt. 

An  Indorsement  is  an  acknowledgment  of  the  receipt  of  a 
partial  payment,  written  on  the  back  of  a  note,  stating  the 
amount  and  the  date  of  payment. 

242.  The  following  is  the  method  of  solving  questions  in 
partial  payments  when  the   note  is  paid  in  full  in  a  year 
or  less : 

What  amount  is  due  Dec.  20,  1895,  on  a  note  for  11600, 
dated  Jan.  14,  1895,  with  interest  at  6%,  on  which  the 
following  payments  are  indorsed : 

Feb.  20,  1895,  1400;  May  5,  1895,  $200.;  Aug.  2,  1895, 
8600. 

The  time  between  Jan.  14  and  Dec.  20  =  11  mo.  6  da.  ($1600). 
The  time  between  Feb.  20  and  Dec.  20  =  10  mo.  ($400). 
The  time  between  May  5  and  Dec.  20  =  7  ino.  15  da.  ($200). 
The  time  between  Aug.  2  and  Dec.  20  =  4  mo.  18  da.  ($600). 
The  amount  of  $  1600  for  11  mo.  6  da.  @  6  %  =  $  1689.60. 
The  amount  of  $400  for  10  mo.  @  6%  =  $420. 
The  amount  of  $200  for  7  mo.  15  da.  @  6  %  =  $207.50. 


INTEREST  255 

The  amount  of  $600  for  4  mo.  18  da.  @  6%  =  $613.80. 
The  total  amount  of  the  payments  Dec.  20  ==  $420  +  $207.50  +  $613.80 
=  $1241.30. 

.-.  the  amount  due  Dec.  20  =  $1689.60  -  $1241.30  =--  $448.30. 

243.  This  solution  is  in  accordance  with  the  Merchants' 
Rule,  which  is  as  follows :  When  the  note  is  paid  in  full  in  a 
year  or  less,  find  the  amount  of  the  note  and  of  each  payment 
at  the  date  of  settlement.  From  the  amount  of  the  note  subtract 
the  sum  of  the  amounts  of  the  payments.  The  difference  thus 
found  is  the  sum  due  on  the  date  of  settlement. 

Exercise  161 

1.  What  is  due  Nov.  1,  1895,  on  a  note  for   $2000,  dated 
Feb.  1,  1895,  with  interest  at  6%,  on  which  the  following  pay- 
ments are  indorsed :  July  1,  1895,  $  600 ;   Sept.  1,  1895,  $  800  ? 

2.  On  a  note  for  $  2400,  dated  Sept.  8,  1892,  and  drawing  6%, 
the   following   payments   were   indorsed :    Oct.    8,    1892,    $  600 ; 
Jan.  8,  1893,  $250;    July  15,  1893,  $850.     What  was  due  on 
the  note  Sept.  8,  1893  ? 

3.  A  note  for  $1500,  dated  April  14,  1894.  writh  interest  at 
5%,   bears   the   following   indorsements:    June   8,   1894,   $450; 
Oct.  29,  1894,  $  750.     What  is  due  April  14,  1895  ? 

4.  On  a  note  for  $  1650,  dated  Oct.  9, 1893,  and  bearing  interest 
at  5|  % ,  the  following  payments  were  made  :  Dec.  5,  1893,  $  240 ; 
Feb.  25,  1894,  $  320 ;   June  30,  1894,  $  300.     What  is  due  Aug. 
15,  1894  ? 

5.  How  much  was  due   on  the   following   note,  on   Dec.  31, 
1889? 

$  950.  NEW  YORK,  Jan.  2,  1889. 

For  value  received,  I  promise  to  pay  James  Brown  or  order,  on 
demand,  nine  hundred  and  fifty  dollars,  with  interest  from  date 

at  6%  per  annum. 

GEORGE  THOMPSON. 


256  ARITHMETIC 

On  this  note  the  following  payments  were  indorsed :  Feb.  22, 
1889,  $225;  May  22,  1889,  $85;  July  21,  1889,  $125;  Sept.  21, 
1889,  $  325. 

244.  The  following  is  the  method  of  solving  questions  in 
partial  payments  when  the  note  runs  longer  than  a  year : 

A  note  for  -$2400,  dated  March  15,  1893,  and  drawing 
interest  at  6%,  had  the  following  payments  indorsed  upon  it: 
June  30,  1893,  1250;  Sept,  12,  1893,  $25;  April  9,  1894, 
1450;  Sept.  14,  1894,  $84.50.  How  much  was  due  on  the 
note  March  29,  1895  ? 


vr. 

mo. 

da. 

yr.     mo. 

da. 

1893 

6 

30 

(June  30) 

1894 

4 

9 

(April  9) 

1893 

3 

15 

(March  15) 

1893 

9 

12 

(Sept.  12) 

3 

15 

($250) 

6 

27 

($450) 

1893 

9 

12 

(Sept.  12) 

1894 

9 

14 

(Sept.  14) 

1893 

6 

30 

(June  30) 

1894 

4 

9 

(April  9) 

2 

12 

($25) 

5 

5 

($84.50) 

yr.    mo. 

da. 

1895    3 

29  (March  29) 

1894    9 

14  (Sept.  14) 

6        15 

The  interest  on  $2400  from  March  15,  1893,  to  June  30,  1893  =  $42. 
The  amount  due  June  30,  1893-$  2400 +  $42 -$250  =  $2192. 
The  interest  on  $2192  from  June  30,  1893,  to  Sept.  12,  1893  =  $26.30. 
The  interest  on  $2192  from  Sept.  12,  1893,  to  April  9,  1894  =  $75.62. 

The  amount  due  April  9,  1894  =  $2192  +  $26.30  +  $  75.62  -$25  -$450  = 
1818.92. 

The  interest  on  $  1818.92  from  April  9,  1894,  to  Sept.  14,  1894  =  $  46.99. 
The  amount  due  Sept.  14,  1894  ==  $1818.92+  $46.99  -  $84.50  =  $1781.41. 
The  interest  on  $  1781.41  from  Sept.  14,  1894,  to  March  29,  1895  ==  $57.90. 
/.  the  amount  due  March  20,  1895  =  $  1781.41  +  $  57.00  =  f  1839.31. 


INTEREST  257 

NOTE.  —  In  the  solution  of  the  above  example,  the  interest  on  $2192  from 
June  30,  1893,  to  Sept.  12,  1893,  is  found  to  be  $26.30,  which  is  greater  than 
the  payment  made  on  Sept.  12  ;  consequently,  the  interest  is  again  computed 
on  $2192  from  Sept.  12,  1893,  to  April  9,  1894,  and  the  interest  is  found  to  be 
$75.62.  The  sum  of  the  two  payments  $25  and  $450  is  now  greater  than 
the  sum  of  the  two  interests  $26.30  and  $  75.62,  and  the  solution  proceeds  as 
given  above. 

245.  The  following  is  the  rule  for  finding  the  amount  due 
on  a  note  at  a  given  date,  when  the  sum  is  not  paid  within  a 
year.  It  is  known  as  the  United  States  Rule  because  it  is 
adopted  by  the  Supreme  Court  of  the  United  States.  Find  the 
amount  of  the  principal  until  the  time  of  a  payment.  Subtract 
the  payment  made  from  this  amount,  and  use  the  remainder  for 
a  new  principal.  Continue  this  process  until  the  time  of  settle- 
ment, ivhen  the  last  amount  is  the  sum  due.  If,  however,  any 
payment  is  less  than  the  accrued  interest,  compute  the  interest 
for  the  next  period  on  the  same  principal  as  before,  and  do  this 
until  the  sum  of  the  payments  equals  or  is  greater  than  the  sum 
of  the  interests. 

Exercise  162 

1.  On  a  note  for  $  2000,  dated  Jan.  24,  1890,  and  drawing  6% 
interest,   are  indorsed  the  following  payments:    May  24,  1890, 
$  440  ;  Aug.  24,  1890,  $  324  ;  Feb.  24,  1891,  $  139.     How  much 
is  due  July  24,  1891  ? 

2.  A  note  for  $  1200,  dated  Jan.  18,  1889,  and  drawing  interest 
at  5%,  had  payments  indorsed  upon  it  as  follows:    March  18, 
1889,  $  210 ;  Sept.  18, 1889,  $  15  ;  Feb.  12, 1890,  $  260.     Find  the 
balance  due  May  9,  1890. 

3.  On  a  mortgage  for  $3750,  dated  May  16,  1887,  and  bearing 
interest  at  6%,  there  were  paid  May  16,  1888,  $350;   Sept.  18, 
1888,  $280;  Jan.  22,  1889,  $750;  May  16,  1889,  $925.     What 
sum  was  due  on  the  mortgage  Oct.  31,  1889  ? 


258  ARITHMETIC 

4.    How  much  was  clue  on  the  following  note,  Oct.  30, 1889  ? 

$  850.  NEW  YORK,  Oct.  30,  1887. 

For  value  received,  I  promise  to  pay  Alex.  Thompson  or  order, 
on  demand,  eight  hundred  and  fifty  dollars,  with  interest  from 

date  at  6%.  TO 

JOHN  STUART. 

On  this  note  the  following  payments  were  indorsed :  April  20, 

1888,  $125;  Nov.  20,  1888,  $125;  Jan.  20,  1889,  $75;  July  20, 

1889,  $  425. 

COMPOUND  INTEREST 

246.  Compound  Interest  is  interest  which  is  found  for  stated 
periods  and  added  at  the  end  of  each  period  to  the  principal, 
the  sum  of  the  principal  and  interest  becoming  the  new  prin- 
cipal. 

The  unit  of  time  is  1  yr.,  although  the  interest  may  be 
compounded  annually,  semiannually,  quarterly,  and  so  on. 

Thus  6%  compounded  semiannually  means  that  each  new 
principal  is  increased  each  6  mo.  by  3%  of  itself. 

247.  If  15000  deposited  at  a  savings  bank  draws  interest 
at  4%,  semiannually,  the  interest  due  at  the  end  of  the  first 
half-year  will  be  2%  of  $5000  or  '$100. 

If  this  $  100  is  not  drawn,  it  is  placed  to  the  credit  of  the 
depositor,  who  has  now  $5100  on  deposit. 

The  interest  for  the  second  half-year  is  2%  of  $5100  or 
8102. 

If  this  is  not  drawn,  it  is  placed  to  the  credit  of  the  depos- 
itor, making  his  deposit  $5202. 

The  interest  for  the  third  half-year  is  2%  of  $5202  or 
$104.04. 

If   this  is  not  drawn,  it  is  placed  to  the   credit  of  the 


INTEREST  259 


depositor,  making  his  deposit  15306.04  at  the  end  of  1  yr. 

6  mo. 

Thus  15000  at  4%  interest,  compounded  semiannually, 
will  in  1  yr.  6  mo.  amount  to  $5306.04;  and  the  compound 
interest  for  that  time  will  be  15306.04  -  $5000  =  $306.04. 


$  5000  original  principal 

1.02 
10000 
50000 


$5100.00  amount  at  the  end  of  the  first  period 

1.02 
10200 
51000 


$5202.00  amount  at  the  end  of  the  second  period 

1.02 
10404 

52020 


$5306.04  amount  at  the  end  of  the  third  period 

248.  Find  the  compound  interest  on  $5000  for  1  yr. 
10  mo.  15  da.  at  4%,  payable  semiannually. 

As  in  the  last  paragraph,  find  the  amount  of  $5000  for 
1  yr.  6  mo.,  and  then  complete  the  work  thus : 

The  rate  per  cent  for  4  mo.  15  da.  or  f  yr.  - :  f  x  4  %  =  H  %. 

$5306.04  amount  at  the  end  of  the  third  period 
1.01} 


265302 
530604 

5306040  » 

$5385.6306  amount  at  the  end  of  the  fourth  period 

/.  the  compound  interest  =  $5385.63  -  $  5000  -  $385.63. 

Instead  of  finding  the  rate  for  4  mo.  15  da.,  we  might  have  found  the 
interest  on  $5306.04  for  1  yr.  at  4%,  and  then  have  taken  f  of  it  to  find 
the  interest  for  the  last  period,  since  4  mo.  15  da.  is  f  of  1  yr. 


260  ARITHMETIC 

249.  Find  what  principal  will  in  1  yr.  10  mo.  and  15  da. 
amount  to  15385.63. 

From  a  study  of  the  preceding  paragraph  it  will  appear 
"that  to  find  the  principal  at  the  beginning  of  the  fourth 
period,  or  the  amount  at  the  end  of  the  third  period,  we 
must  divide  $5385.63  by  1.01  J  or  1.015,  since  the  fourth 
amount  is  found  by  multiplying  the  fourth  principal  by 
1.011 

The  principal  at  the  beginning  of  the  fourth  period  =  .$  5385.63  -~  1.015 

=  $5306.04. 
The  principal  at  the  beginning  of  the  third  period     =  $  5306.04  -^  1.02 

=  $5202. 
The  principal  at  the  beginning  of  the  second  period  =  $5202  -r-  1.02 

=  $  5100. 
The  principal  at  the  beginning  of  the  first  period       =  $  5100  -=-  1.02 

=  $  5000. 
.-.  the  principal  =  $  5000. 

In  practice  it  is  often  more  convenient  to  divide  by  the  1.02  three  times 
in  succession  and  then  by  1.015  for  the  last  division. 

Exercise  163 

Find  the  amount  and  the  compound  interest  of : 

1.  $  800  for  3  yr.  at  5%,  compounded  annually. 

2.  $425  for  4  yr.  at  4%,  compounded  annually. 

3.  $250  for  2  yr.  at  6%,  compounded  semiannually. 

4.  Find  the  amount  and  also  the  compound  interest  on  $1000 
for  3  yr.  at  5%.  + 

5.  In  question  4  what  would  the  amount  have  been  at  simple 
interest  ?     How  much  has  to  be  paid  as   interest  on    interest  ? 
What  fraction  is  it  of  the  first  year's  interest  ?    What  per  cent  ? 

6.  Find  the  amount  of  $300  for  2  yr.  at  G%,  interest  payable 
semiannually. 


INTEREST  261 

7.  Find  the  amount  of  $650  for  1  yr.  3  mo.,  interest  payable 
quarterly  at  4%  per  annum. 

8.  Find  the  compound  interest  on  $8240   for  2  yr.  at  5%, 
payable  semiannually. 

9.  State  how  to  find  the  amount  of  a  sum  of  money  at  com- 
pound interest,  for  a  given  time  and  rate. 

10.  Find  the  amount  and  also  the  compound  interest  on  $2500 
for  1  yr.  10  mo.  and  15  da.  at  6%,  payable  semiannftally. 

11.  What  principal  will  amount  to  $  2247.20  in  2  yr.  at  6%? 

12.  What  sum  of  money  put  out  at  compound  interest  for  2  yr. 
at  7%  will  amount  to  $100? 

13.  What  sum  of  money  put  out  for  2  yr.  at  5%,  payable  half- 
yearly,  will  amount  to  $  600  ? 

14.  State  how  to  find  the  principal  that  amounts  to  a  given 
sum  of  money  at  compound  interest  for  a  given  time  and  rate. 

15.  Find   the  difference   between   the   simple   and  compound 
interest  of  $  1050  for  3  yr.  at  4%. 

16.  A  sum  of  money  put  out  at  simple  interest  for  2  yr.  at  6% 
amounted  to  $  896.     To  what  sum  would  it  have  amounted  had  it 
been  lent  at  compound  interest  ? 

17.  The  simple  interest  on  a  sum  of  money  for  3  yr.  at  7%  is 
$  420.     What  is  the  compound  interest  of  the  same  sum  for  the 
same  time  ? 

18.  A  man  deposits  in  the  savings  bank  $  1500,  on  which  the 
interest  at  3%  per  annum  is  to  be  added  to  the  principal  every 
6  mo.     How  much  money  has  the  man  in  the  bank  at  the  end 
of  2  yr.  ? 

19.  What  will  be  the  amount,  compound  interest,  of  $  2400  for 
1|-  yr.  at  10%  per  annum,  paid  half-yearly,  and  at  what  rate,  simple 
interest,  will  it  amount  to  the  same  sum  in  the  same  time  ? 


262  ARITHMETIC 

*  ANNUAL  INTEREST 

250.  Annual  Interest  is  the  sum  of  the  simple  interest  on 
the  principal  and  on  each  year's  interest,  if  unpaid,  from  the 
time  it  is  due  until  the  date  of  settlement. 

Annual  Interest  is  charged  when  the  words  "  interest  pay- 
able annually  "  are  in  the  note. 

251.  Find   the   amount  due   Oct.  9,  1895,  on   a  note  for 
11200,  dated  July  5,  1891,  with  interest  payable  annually  at 
6%. 

yr.  mo.       da. 

(1)  1895        10        9 

1891          7        5 

434 

The  interest  on  $  1200  for  1  yr.  at  6  %  =  $  72. 
The  interest  on  $  1200  for  4  yr.  3  ino.  4  da.  at  6  %  =  $  306.80. 

(2)  The  interest  due  July  5,  1892,  bears  interest  for  3  yr.  3  mo.  4  da. 
The  interest  due  July  5,  1893,  bears  interest  for  2  yr.  3  mo.  4  da. 
The  interest  due  July  5,  1894,  bears  interest  for  1  yr.  3  mo.  4  da. 
The  interest  due  July  5,  1895,  bears  interest  for  3  mo.  4  da. 

.  •.  the  interest  on  $  1200  for  1  yr.,  i.e.  $  72,  bears  interest  for  7  yr.  16  da. 

The  interest  on  $  72  for  7  yr.  16  da.  @  6  %  =  $  30.43. 
.-.  the  amount  due  =  $  1200  +  $  306.80  +  $  30.43  =  $  1537.23. 

Exercise  164 

1.  Find  the  amount  due  July  16,  1896,  on  a  note  for  $  900, 
dated  July  16,  1892,  with  interest  payable  annually  at  6%. 

2.  Find  the  amount  due  March  9,  1896,  on  a  note  dated  Jan.  3, 
1892,  for  $  1500  at  5%,  interest  payable  annually. 

3.  Find  the  amount  clue  Sept.  26,  1896,  on  a  note  for  $  280, 
dated  June  5,  1893,  with  interest  at  41%,  payable  annually. 

4.  Find  the  amount  due  June  8, 1896,  on  a  note  dated  Aug.  12, 
1892,  for  $  712.50,  with  interest  at  6%,  payable  annually. 


INTEREST  263 

5.  Find  the  amount  due  Nov.  4,  1896,  on  a  note  dated  Dec.  24, 
1892,  for  $842,  with  interest  at  4-}%,  payable  annually. 

6.  Find  the  simple,  annual,  and  compound  interest  on  a  note 
for  $  1000,  dated  Aug.  12,  1892,  and  due  Aug.  12,  1896,  interest 
at  6%. 

7.  Compare  the  methods  of   finding  the  simple,  annual,  and 
compound  interest  on  a  sum  of   money,  for  a  given  time,  at  a 
given  rate  per  cent. 

*  STOCKS  AND  BONDS 

252.  The   capital  of   a  bank  or  other  public  company  is 
called  Stock. 

It  is  usually  divided  into  a  definite  number  of  equal  parts 
or  Shares. 

The  original  value  of  a  share,  generally  $100,  850,  or  $25, 
is  called  its  Par  Value. 

253.  The  Market  Value  of  a  share  is  the  sum  for  which  it 
can  be  sold. 

Stock  is  said  to  be  above  par,  or  at  a  premium,  when  the 
market  value  is  greater  than  its  par  value;  it  is  said  to  be 
below  par,  or  at  a  discount,  when  the  market  value  of  the 
share  is  less  than  its  par  value. 

Thus  if  ^100  stock  sells  for  $112  money,  the  stock  is  at 
12%  premium,  and  it  is  said  to  sell  at  112. 

If  1100  stock  sells  for  896  money,  the  stock  is  at  4%  dis- 
count, and  is  quoted  at  96. 

254.  A  Stock  Broker  is  a  person  who  buys  or  sells  stocks, 
bonds,  or  similar  securities.    His  commission,  called  Brokerage, 
is  reckoned  at  a  certain  rate  per  cent,  which  varies,  the  most 
common  rate  being  l  of  1%  or  -|%. 


264  ARITHMETIC 

255.  A  Bond  is  a  note  bearing  interest  issued  by  a  govern- 
ment  or   corporation.      There    are    two    kinds   of   bonds,  - 
registered  and  coupon  bonds. 

A  Registered  Stock  or  Bond  is  one  which  is  registered  on 
the  books  of  the  company  or  government  issuing  it,  and 
which  cannot  be  sold  or  transferred  except  in  writing  at  the 
office  of  the  treasurer. 

An  Interest  Coupon  is  an  interest  certificate  payable  to  the 
bearer,  which  is  attached  to  the  bond,  and  which  is  detached 
when  the  interest  becomes  due. 

One  coupon  is  attached  to  the  bond  for  each  instalment 
of  interest  to  be  paid  on  it. 

256.  The  following  is  the  quotation  of  U.  S.  bonds  in  the 
market  of  July  8,  1896 : 

Bid  Asked 

Registered  2's  95  ... 

Registered  4's  108  108| 

Coupon  4's  109  109£ 

New  Coupon  4's  116|  116f 

Registered  5's  112f  113 

257.  The  following  is  the  quotation  of  stock  in  the  market 
of  July  8,  1896  : 

Closing 
Stocks  Opening  Highest  Lowest  July  8  July  7 

Am.  Sugar  109£  llli  109f  110  110£ 

Am.  Sugar  pfd.  101 1  lOlf  lOlf  lOlf  101  £ 

C.  B.  &  Q.  71|  72f  71f  72|  72| 

C.  R,  I.  &  P.  63  63|  62f  62|  63^ 

Michigan  Central          96  96  96  96 

Manhattan  97J  97£  961  96|  97 

Del.  &  Hud.  1241  1241  1241  1241  1241 

Exercise  165 

1.  At  what  different  prices  is  Am.  Sugar  stock  quoted  at, 
July  8,  1896  ? 


INTEREST  265 

2.  What  will  a  seller  receive  from  his  broker  for  1  share  of 
C.  B.  &  Q.  stock,  July  8,  1896,  at  each  of  the  quoted  prices, 
brokerage  being  J-%?     What  from  1  share  of  Am.  Sugar  pfd.  ? 

3.  What  will  a  buyer  have  to  pay  for  1  share  of  Manhattan 
stock  at  each   quotation,  July  8,  1896,  brokerage  'J%?      What 
for  C.  R.  I.  &  P.  ? 

4.  At  what  per  cent  premium  are  the  different  quotations  for 
Am.  Sugar,  Ain.  Sugar  pfd.,  and  Del.  &  Hud.  stock,  July  8, 1896  ? 

5.  At  what  per  cent  discount  are  the  different  quotations  for 
C.  B.  &  Q.,  C.  R.  I.  &  P.,  Michigan  Central,  and  Manhattan  stock, 
July  8,  1896  ? 

6.  What  would  I  receive  for  1  share  of  Del.  &  Hud.,  July 
8,  1896,  sold  at  the  highest  price,  brokerage  i%?      What  for  10 
shares  ?     What  for  100  shares  ? 

7.  What  would   I   have   to   pay  for   1    share  of  C.  B.  &  Q. 
stock,  July  8,  1896,  bought  at  the  opening  price,  brokerage  J%? 
What  for  10  shares  ?     What  for  100  shares  ? 

8.  What  would  I   have  to   pay  for   1    share  of   Am.  Sugar 
stock,  July  8,  1896,  at  the  lowest  quoted  price,  brokerage  |%? 
What  for  10  shares  ?     What  for  100  shares  ? 

9.  What  would  I  receive  for  1  share  of   C.  B.  &  Q.  R.  R. 
stock,  July  8,  1896,  sold  at  the  lowest  quotation,  brokerage  -§-%? 
What  for  10  shares  ?     What  for  100  shares  ? 

10.  What  will  1  share  of  C.  B.  &  Q.  stock  cost,  July  8,  1896, 
at  the  opening  price,  brokerage  -§-%?      How  many  shares  can  I 
buy  for  $144?     For  $  216?     For  $360? 

11.  What  will   1    share  of   C.  R.  I.  &  P.  stock  cost,  July  8, 
1896,  at  the  lowest  quotation,  brokerage  ^%?      How  many  shares 
can  be  bought  for  f  125  ?     For  $  625  ? 

12.  What  is  the   difference  between  the  highest  and  lowest 
quotations  of  Manhattan  stock,  July  8,  1896  ? 


266  ARITHMETIC 

13.  What  is  the  difference  between  the  closing  prices  of  Del.  & 
Hud.  stock,  July  7  and  July  8,  1896  ? 

14.  What  is  the  difference  between  the  opening  and  closing 
prices  of  C.  B.  &  Q.  stock,  July  8,  1896  ? 

258.    (1)  How  much  will  be  realized  by  selling  out  66 
shares  of  N.  Y.  Central  R.  R.  stock  at  95  J,  brokerage  J%? 

1  share  of  stock  sells  for  $95|  --  $  i  or  $95f  money. 
.-.  66  shares  of  stock  sell  for  66  x  $95f  or  .$6294.75  money. 

NOTE.  —  The  brokerage  =  66  x  $  £  =  $8.25. 

(2)  How   many   shares   of   Del.  &    Hud.  stock  at   124|, 
brokerage  1%,  can  I  buy  for  $5591.25? 

1  share  costs  $  124-J-  +  $  1  or  $  124.25. 
.-.  the  number  of  shares  =  $  5591.21  ~  $  124.25  —  45. 

NOTE.  —The  brokerage  =  45  x  $  |  =  $ 5.62^. 

(3)  A  broker  realizes  §7.25  from  a  sale  of  stock,  brokerage 
|%.     What  was  the  par  value  of  the  stock  sold? 

i  %  of  the  par  value  =  $  7.25. 
1  %  of  the  par  value  =  $  58. 
100  %  of  the  par  value  =  $  5800. 
.-.  the  par  value  =  $  5800. 

(4)  I  sold  through  my  broker  95  shares  of  Chicago  and 
Northwestern  R.  R,  stock,  receiving  for  it  $  9476.25,  brokerage 
\%.     Find  at  what  price  the  stock  was  quoted. 

95  shares  sell  for  $9476.25. 

1  share  sells  for  $  9476.25  -f-  95  =  $  99.75  ; 
i.e.  excluding  brokerage,  the  selling  price  =  99|. 
.-.  stock  is  quoted  at  99|  +  |  or  99|. 

(5)  What  annual  income  will  be  realized  from  $ 3828.12 J, 
invested  in  the  U.  S.  4's  at  109J,  brokerage  J%? 

1  share  costs  $  109}  +  $  \  -.--  $  109|  =  $  109.375. 

The  number  of  shares  =  $3828.125  -f-  $109.375  =  35. 
.-.  the  income  =  35  x  $4  =  $  140. 


INTEREST  267 

(6)  What  amount  of  money  must  be  invested  in  6%  stock, 
at  119J,  brokerage  |%,  to  realize  an  income  of  1978? 

1  share  yields  an  income  of  $  6. 
The  number  of  shares  =  $  978  -=-  $  6  =  163. 
1  share  costs  $  119f  +  $  |  =  $  119|. 
163  shares  cost  163  x  $  119f  =  $  19,539.621. 
.-.  $  19,539.62i  must  be  invested. 

(7)  If  6%  stock  is  bought  at  109  J,  what  per  cent  does  it 
pay  on  the  investment,  brokerage  J  %  ? 

1  share  costs  $  109 1  +  $  |  =  $  110. 

$  110  yields  an  income  of  $  6. 

.-.  the  rate  per  cent  =  Tf  „  or  5T5T%  of  the  investment. 

(8)  What  must  I  pay  for  6%  stock  to  realize  an  income  of 
on  the  investment,  brokerage  \ 


8  %  of  the  cost  of  1  share  = 
1  %  of  the  cost  of  1  share  =  $  j. 
100  %  of  the  cost  of  1  share  ==  $75  ; 
i.e.  including  brokerage,  the  cost  price  is  $  75. 
/.  stock  is  quoted  at  74|. 

Exercise  166 

1.  What  will  25  shares  of  Adams  Express  stock  cost  at  148, 
brokerage  %%? 

2.  What  is  realized  from  the  sale  of  208  shares  of  C.  B.  &  Q. 
R.  R.  stock  at  71 J,  brokerage  J-%? 

3.  What  did  I  pay  for  39  shares  Chicago  and  Northwestern, 
July  8,  1896,  stock  selling  at  99J  and  brokerage  being  i%? 

4.  Find  what  I  received  from  the  sale  of  84  shares  of  Western 
Union  stock  at  821,  brokerage  i%. 

5.  What  is  the  cost  of  $20,000  U.   S.  4's  at  112f,   broker- 
age J%? 

6.  Find  the  cost  of   f  24,0()0   U.   S.  4's  at   116f,   brokerage 


268  ARITHMETIC 

7.  July  8,  1896,  40  shares  of  Chicago  City  Railway,  reg.  at 
220,  were  sold,  brokerage  ^%.     Find  what  was  received  by  the 
owner  of  the  stock. 

8.  How  many  shares  of  Manhattan  R.  R.  stock  at  97^  can  I 
buy  for  $3505.50,  brokerage  $%? 

9.  July  8,  1896,  Am.  Sugar  pfd.  stock  was  quoted  at  101 J. 
How  many  shares  were  bought  for  $4257.75,  brokerage  £%? 

10.  A  stockholder  sold  D.  L.  and  W.  R.  R.  stock  at  157 J,  receiv- 
ing all  together  $  3771.     How  many  shares  did  he  sell,  brokerage 
being  ±%? 

11.  How  many  shares  of  Wells  Fargo  Express  stock  must  I 
sell  at  95,  brokerage  £%,  to  receive  $9677.25? 

12.  If  from  my  sales  of  New  York  Central  R.  R.  stock  at  95^,  I 
receive  $6278.25,  how  much  stock  did  I  sell,  brokerage  being ^%? 

13.  How  many  shares  of  D.  L.  and  W.  R.  R.  stock  at  158J  can 
be  bought  for  $2855.25,  brokerage  £%? 

14.  A  broker  sells  24  shares  of  stock  on  a  commission  of  \%. 
How  much  does  he  realize  ? 

15.  How  many  shares  of  stock  does  a  broker  sell  to  realize  a 
commission  of  $16.25,  brokerage  i%? 

16.  A  broker  realizes  $12.50  from  the  sale  of  stock,  brokerage 
1%.     What  was  the  par  value  of  the  stock  sold  and  what  did  it 
sell  for  at  70|  ? 

17.  A  broker  received  $46.50  for  buying  stock  on  a  commis- 
sion of  f  °Jo'     How  much  stock  did  he  buy  ? 

18.  I  sold  through  my  broker  40  shares  of  stock,  receiving  for 
it  $4860,  brokerage  1%.     At  what  price  was  the  stock  quoted  ? 

19.  A  person  received  $  6053.12^-  for  $  6500  stock  after  paying 
his  broker  i%.    Find  at  what  per  cent  discount  the  stock  was  sold. 

20.  July  7,  1896,  $  1654.25  was  paid  for  26  shares  of  Rock 
Island  R.  R.  stock,  brokerage  i%.     At  what  was  Rock  Island 
stock  quoted,  July  7  ? 


INTEREST  -269 

21.  What  annual  income  will  be  obtained  from  $  6071,  invested 
in  U.  S.  4's  coup,  of  1925  at  116f,  brokerage  i%? 

22.  A  person  paid  $8578.50  for  U.  S.  4's  at  112f,  brokerage 
\%.     What  was  his  income  from  the  bonds  ? 

23.  If  I  invest  $8583.75  in  stock  at  95£,  paying  5%  dividend, 
what  will  be  my  income,  brokerage  ^%? 

24.  What  income  will  be  realized  from  $  9229.50  invested  in 
stock  at  109|,  brokerage  \°/b,  paying  a  dividend  of  5|%? 

25.  What  amount  of  money  must  be  invested  in  8%  stock  at 
158^-,  brokerage  i-%,  to  realize  an  income  of  $  1096  ? 

26.  What  sum  must  I  invest  in  4|%  stock  at  99|,  to  produce 
an  annual  income  of  $  1638,  brokerage  |- 


27.    How  much  must  I  invest  in  U.  S.  5's  at  112|  to  realize 


an  annual  income  of  $450,  brokerage 

28.  If  street  railway  stock  bought  at  232  yields  a  half-yearly 
dividend  of  6£%,  how  much  must  I  invest  to  obtain  a  semiannual 
income  of  $325,  brokerage  |%? 

29.  If  I  buy  stock  through  a  broker  who  charges  J-%,  how 
much  must  I  invest  in  stock  at  153,  paying  9%  dividends,  to 
secure  an  income  of  $  1350  ? 

30.  If  4j-%  stock  is  bought  at  74J,  brokerage  i%,  what  per 
cent  does  it  pay  on  the  investment  ? 

31.  If  8%  stock  is  bought  at  159J,  what  per  cent  does  it  pay 
on  the  investment,  brokerage  -J-%? 

32.  What  must  I  pay  for  4%  stock  to  pay  5%  on  the  invest- 
ment, brokerage  i%? 

33.  What  rate  of  interest  do  I  realize  on  an  investment  in  6% 
stock  at  107  J,  brokerage  i%? 

34.  What  must  I  pay  for  5%  stock  to  yield  an  income  of  6% 
on  my  investment  ? 

35.  A  person  receives  $,600  from  an  8%  bank  dividend.    How 
much  stock  does  he  own  ? 


270'  ARITHMETIC 

36.  A  person  having  $5000  bank  stock  sells  out  when  it  is  at 
40  %  premium.    What  amount  of  money  does  he  receive,  brokerage 
being  %%? 

37.  Bought  through  a  broker  1600  shares  ($  100)  R.  R.  stock  at 
69^,  brokerage  £%.     What  was  the  gross  cost  of  the  stock  ? 

38.  A  speculator  bought  36,500  shares  ($100)  R.  R.  stock  at 
39f,  and  sold  them  at  40f.     What  was  his  gain,  brokerage  i-%  on 
both  transactions  ? 

39.  A  bank  declared  a  dividend  of  3^%.     How  much  should  a 
stockholder  owning  120  shares  ($  50)  receive  ? 

40.  One  company  guarantees  to  pay  6%  on  shares  of  $100 
each;  another  guarantees  at  the  rate  of  5-f%  on  shares  of  $30 
each ;  the  price  of  the  former  is  $  124.50,  and  of  the  latter  $  34. 
Find  the  rates  of  interest  which  they  return  to  the  purchaser. 

41.  A  broker  receives  $42,100  to  invest  in  U.S.  5-20  bonds, 
after  reserving  J%  on  the  par  value  of  the  amount  purchased. 
What  was  his  commission,  the  bonds  being  at  a  premium  of  5%? 

42.  A  man  bought  through  a  broker  1900  shares  ($  100)  R.  R. 
stock  at  54J  and  sold  them  at  55-f .     What  was  his  net  profit  on 
the  transaction,  brokerage  each  way  %%'! 

43.  An  insurance  company  declared  a  dividend  of  9%.     What 
rate  is  that  on  the  market  value  of  the  shares  which  are  at  185  ? 

44.  Compare  the  rates  on  the  cash  values  of  6%  on  stock  at 
216  and  3^%  on  stock  at  125. 

45.  Sold  37  shares  ($  25)  B.  and  L.  Association  stock,  receiving 
therefor  $  1019.81.     At  what  rate  was  the  stock  sold  ? 

46.  Bought  through  a  broker  750  shares  ($50)  in  the  Farmers' 
Loan  and  Savings  Society,  paying  therefor  $  43,968.75.    At  what 
quotation  were  they  bought,  brokerage  J-%? 

47.  Bought  stock  at  197f  and  sold  it  at  194J-,  having  mean- 
while received  a  dividend  of  6%  on  it.     My  net  gain  on  the 
transaction  after  paying  1%  brokerage  each  way  is  $336.     How 
many  shares  ($  40)  did  I  buy  ? 


INTEREST  271 

48.  How  many  railway  shares  ($100)  at  40%  discount  must 
be  sold  in  order  that  the  proceeds  invested  in  bank  stock,  which 
is  4%  below  par,  and  pays  a  dividend  of  7%,  may  yield  an  income 
of  $  1680,  brokerage  included  ? 

49.  Explain  the  terms :  Stocks,  Shares,  Dividends.     When  is 
stock  at  par  ?     At  a  premium  ?     At  a  discount  ? 

50.  When  the  3J  per  cents  are  at  98,  what  must  be  the  price 
of  another  stock  yielding  4-J%,  so  that  the  latter  may  be  as  profit- 
able as  the  former,  brokerage  included? 

EXCHANGE 

259.  If  A  of  Chicago  owes  B  of  St.  Paul  a  sum  of  money, 
he  can  discharge  the  debt  in  any  one  of  several  ways.     He 
can  buy  a  post-office  order  at  the  Chicago  post-office  payable 
to  B  at  the  post-office  at  St.  Paul ;   he  can  buy  an  express 
order  at  the  office  of  an  express  company,  payable  to  B  at 
any  office  of  the  same  company ;  or  he  can  buy  a  draft  at  a 
bank  payable  to  B  at  a  bank  in  St.  Paul. 

Give  some  reasons  why  it  is  better  to  discharge  a  debt  by 
means  of  a  post-office  order,  express  order,  or  draft  than  by 
sending  the  money  in  a  registered  letter  or  by  express  or 
check. 

260.  The  following  are  the  rates  charged  for  express  orders 
to  any  part  of  the  United  States  or  Canada : 

Rates  for  orders  not  over 

$5.00,  5£  $40.00,  18  £ 

10.00,  8£  60.00,  20^. 

20.00,  10  £  75.00,  25  £ 

30.00,  15  £  100.00, 

Over  $  100  at  above  rate. 


272  ARITHMETIC 

261.  The  rates  charged  for  post-office  orders  to  any  part  of 
the  United  States  are  the  same  as  for  express  orders  up  to 
$100,  but  orders  not  exceeding  $2.50  are  sold  at  a  charge  of 
3^.  Single  post-office  orders  are  not  issued  for  more  than 
$  100,  and  for  larger  amounts  additional  orders  are  issued. 

Exercise  167 

1.  What   is  the  cost  of   an  express  order  for   $25?     $44? 
$  73  ?     $  78  ? 

2.  What  is  the  cost  of  a  post-office  order  for  $80?     $82? 
$  95  ?     $  1.50  ? 

3.  What  is  the  cost  of  an  express  order  for  $  75  ?     $  100  ? 

4.  What  is  the  cost  of  a  post-office  order  for  $  75  ?     $  100  ? 

5.  What   is  the  cost  of  a  draft   for   $75?     $100?     $150? 
$240?     $325?     $180?     The  charge  in  each  case  is  \%   and 
the  least  charge  25  £ 

6.  By  which  of  the  three  methods  given  in  questions  3,  4?  and  5 
is  it  cheaper  to  send  money  in  sums  greater  than  $  75  ?     In  sums 
less  than  $  75,  if  25  ^  is  the  smallest  charge  for  a  draft  ? 


262.  Exchange  is  generally  conducted  through  bankers, 
who  issue  drafts  directing  a  second  bank  to  pay  a  specified 
sum  of  money  to  the  order  of  the  person  named  in  the  draft. 

A  Time  Draft  is  one  payable  at  a  specified  time  after  sight 
or  date. 

If  A  in  Chicago  owe  B  in  St.  Paul  a  sura  of  money,  B  may  send  a  draft  to 
A  for  the  amount.  If  A  accepts  the  draft,  he  writes  the  word  "accepted'' 
with  the  date  across  the  face  and  signs  his  name. 

Exchange  is  the  system  of  paying  debts  to  persons  in  dis- 
tant places  without  actually  sending  the  money,  by  means 
of  money  orders  and  drafts. 


INTEREST  273 

263.  (1)  Find  the  cost  of  a  draft  on  New  York  for  $  600, 
when  exchange  is  \%  premium. 

The  premium  =  \  %  of  $600  =  $  1.50. 
.-.  the  cost  =  $600 +  $1.50  =  $601.50. 

(2)  Find  the  cost  of  a  draft  on  New  Orleans  for  11200, 
payable  60  da.  after  date,  exchange  being  -^%  discount,  and 
interest  6%. 

The  discount  =  1  %  of  $  1200  =  $  3.00. 
The  discount  for  63  da.  =  6  %  of  $  1200  for  63  da.  =  $  12.60. 

.-.  the  cost  =  $  1200  --  $  3.00  --  $  12.60  ==  $  1184.40. 

Show  that  if  exchange  had  been  *-  %  premium,  the  cost  would  have  been 
$1190.40. 

Exercise  168 

1.  Find  the  cost  of  a  draft  for  $  900  at  \°/0  premium. 

2.  Find  the  cost  of  a  draft  for  $  1600  at  \%  discount. 

3.  Find  the  cost  of  a  draft  for  $  4500  at  f  %  discount. 

4.  Find  the  cost  of  a  draft  for  $  2800  at  f  of  1%  premium. 

5.  Find   the  cost  of  a  draft  for  $  1000,  payable  in  60  da., 
exchange  being  \°/0  premium,  and  interest  6%. 

6.  Find   the  cost  of   a  draft   for  $  360,  payable  in  30  da., 
exchange  being  \°/0  discount,  and  interest  5%. 

7.  Find  the  cost  of  a  draft  for  $  1250,  payable  in  60  da., 
exchange  being  \%  premium,  and  interest  4^%. 

8.  Find  the  cost  of  a  draft  for  $  1800,  payable  in  30  da.,  when 
exchange  is  at  par,  and  interest  4%. 

9.  Find  the  cost  of  a  bill  of  exchange  on  London  for  £  600, 
when  exchange  is  quoted  at  $  4.88. 

10.  Find  the  cost  of  a  60-da.  draft  on  Liverpool  for  £  750, 
exchange  at  60  da.  being  $  4.86. 

11.  What  will  be  the  cost  of  a  bill  of  exchange  in  Paris  for 
2400  francs  at  5.16J  francs  per  $  1  ? 

12.  What  will  be  the  cost  of  a  bill  of  exchange  on  Berlin,  for 
2400  marks,  the  rate  of  exchange  being  95^  ct.  for  4  marks  ? 


CHAPTER   XV 

KATIO  AND  PROPORTION 

264.  If  two  quantities  be  expressed  in  terms  of  the  same 
unit,  their  Ratio  is  the  quotient  obtained  by  dividing  the  num- 
ber measuring  the  first  quantity  by  the  number  measuring 
the  second  quantity. 

Thus  the  ratio  of  $  3  to  $  6  =  -|,  or,  as  it  is  frequently 
written,  3:5. 

The  first  term  of  a  ratio  is  called  the  Antecedent,  and  the 
second  the  Consequent. 

Since  a  ratio  may  be  expressed  as  a  fraction,  both  terms 
of  a  ratio  may  be  multiplied  or  divided  by  the  same  number 
without  changing  its  value. 

265.  (1)  If  15  bbl.  of  flour  cost  I  111,  what  will  35  bbl.  cost  ? 
Multiply  $  111  by  ff .     v  35  bbl.  will  cost  f  f  of  what  15  bbl.  cost. 

rf    (fljlll      v     35     <t  9P>Q 

jj  —  iff  ~Y~   •*•  Yl>  —  sP  ^Ut/. 

Or  the  question  may  be  solved  thus : 

15  bbl.  cost    $111. 
1  bbl.  costs  $  -W-. 


.-.  35  bbl.  cost    35  x  *»LiL  =  $259. 

15 

(2)  If  56  men  do  a  piece  of  work  in  21  da.,  how  long  will 
24  men  require  to  do  it? 

Multiply  21  da.  by  i>f.     v  24  men  will  take  |f  as  long  as  56  men, 

da. 

x  =  ^-  x  f  f  =  49  da. 

274 


RATIO   AND   PROPORTION  275 

Or  thus : 

56  men  do  the  work  in  21  da. 

1  man  can  do  the  work  in  56  x  21  da. 

56  x  21 
.*.  24  men  can  do  the  work  in or  49  da. 

24 

Exercise  169 

1.  If  6  articles  cost  $  14.30,  how  much  will  13  cost  at  the 
same  rate  ? 

2.  If  25  Ib.  of  tea  cost  $  16,  how  many  Ib.  can  be  bought  for 

$56? 

3.  If  the  4-lb.  loaf  costs  11  $  when  flour  is  $  6  a  bbl,  find 
its  cost  when  flour  is  $  1\  a  bbl. 

4.  A  bankrupt  owes  $  3000 ;   his  assets  are  $  1740.      What 
sum  will  a  creditor  receive  whose  claim  is  $  350  ? 

5.  The  expense  of  carpeting  a  room  was  $  100;  if  the  breadth 
of  the  room  had  been  4  ft.  greater,  the  expense  would  have  been 
$  120.     Find  the  breadth. 

6.  If  a  man  working  9-|  hr.  per  da.  finishes  a  piece  of  work  in 
6  da.,  in  what  time  would  he  have  finished  it  if  he  had  worked 
8J  hr.  per  da.  ? 

7.  If  a  garrison  of  1500  men  have  provisions  for  13  mo.,  how 
long  will  their  provisions  last  if  it  be  increased  to  2200  ? 

8.  If  4  men  or  6  women  can  do  a  piece  of  work  in  20  da.,  how 
long  will  it  take  3  men  and  15  women  to  do  the  same  work  ? 

9.  A  creditor  receives  $  1.50  for  every  $  4  of  what  was  due 
to  him,  and  thereby  loses  $  301.05.     What  was  the  sum  due  ? 

10.  In  a  certain  business  one  partner,  whose  share  is  f\  of  the 
whole,  receives  from  it  a  profit  of  $  859.20.    'What  share  is  owned 
by  another,  whose  profit  is  $  1969  ? 

11.  A  person  contracts  to  do  a  piece  of  work  in  30  da.,  and 
employs  15  men  upon  it ;  the  work  is  half  finished  in  24  da.    How 


276  ARITHMETIC 

many  additional  workmen  must  be  then  introduced  in  order  to 
perform  the  contract  ? 

12.  The  profits  of  a  garden  for  2  yr.  were  $  1456;  the  profits 
of  the  second  yr.  being  ||-  of  those  of  the  first.     Find  the  profits 
of  each  yr. 

13.  If  10  men  can  do  a  piece  of  work  in  12  da.,  how  soon  after 
beginning  must  they  be  joined  by  3  more  so  as  to  finish  the  work 
in  10  da.  ? 

14.  If  $  120  gain  $  5.81  in  126  da.,  find  the  gain  in  360  da. 

15.  A  bankrupt  who   is    paying  37 ^  f  on  the    dollar    divides 
among  his  creditors  $  6300.     What  do  his  debts  amount  to? 

16.  If  3  men  or  5  boys  can  do  a  piece  of  work  in  17  da.,  in 
how  many  da.  will  5  men  and  3  boys  do  a  piece  of  work  3  times 
as  great  ? 

17.  If  3  men  can  do  as  much  work  in  a  da.  as  4  boys,  how  long 
will  it  take  64  boys  to  finish  a  piece  of  work  of  which  12  men 
have  done  \  in  16  da.  ? 

18.  If  a  debt  after  a  deduction  of  3%  becomes  $  1008.80,  what 
would  it  have  become  after  a  deduction  of  4%  had  been  made  ? 

19.  Six  sheets  of  paper  measuring  8  in.  by  10  in.  weigh  an 
ounce.     Find  the  weight  of  120  sheets  of  the  same  kind  of  paper, 
each  sheet  measuring  11  in.  by  17  in. 

20.  A  person  walks  from  his  house  to  his  office  at  the  rate  of 
4  mi.  per  hr. ;  but  finding  he  has  forgotten  something,  returns  at 
the  rate  of  5  mi.  per  hr.     Compare  the  time  spent  in  going  with 
that  spent  in  returning. 

21.  One  train  travels  81  mi.  in  20  min.,  and  a  second  train  9  mi. 
in  15  min.     Compare  their  rates  per  hr. 

22.  A  man  can  row  6  mi.  an  hr.  in  still  water.     Compare  his 
rate  of  rowing  down  a  stream  which  flows  at  the  rate. of  21  mi.  an 
hr.  with  his  rate  of  rowing  up. 


RATIO    AND    PROPORTION  277 

23.  One  water  pipe  discharges  141  gal.  per  hr.,  another  dis- 
charges 235  gal.  per  hr.    Compare  their  rates  of  discharge  (a)  per 
hr. ;   (b)  per  min. ;   (c)  per  sec. ;  (d)  per  da. ;   (e)  per  seventh  of 
a  da.     Also  compare  the  times  in  which  the  pipes  would  each 
discharge  (a)  705  gal. ;   (6)  705  qt. ;  (c)  705  pt. 

24.  Two  taps  when  both  open  discharge  water  at  the  rate  of 
481  gal.  per  hr. ;  the  discharge  of  the  smaller  of  the  two  being 
at  the  rate  of  148  gal.  per  hr.     Compare  the  volume  discharged 
by  the  larger  tap  in  any  given  time  with  the  volume  discharged 
by  the  smaller  tap  in  the  same  time.     Compare  also  the  time  in 
which  the  larger  tap  will  discharge  a  given  number  of  gal.  with 
the  time  required  by  the  smaller  to  discharge  the  same  number 
of  gal. 

25.  A  greyhound  pursuing  a  hare  takes  3  leaps  to  every  4  the 
hare  takes ;  but  2  leaps  of  the  hound  are  equal  in  length  to  3 
leaps  of  the  hare.     Compare  the  speed  of  the  hound  with  that  of 
the  hare. 

26.  Milk  is  worth  20^  a  gal.,  but  by  watering  it  the  value  is 
reduced  to  15^  a  gal.     Find  the  proportion  of  water  to  milk  in 
the  mixture. 

27.  Two  men  receive  $  15  for  doing  a  certain  piece  of  work. 
Now  one  man  had  worked  but  3  da.  while  the  other  had  worked  5 
da.  on  the  job.     If  the  money  is  to  be  divided  in  proportion  to  the 
lengths  of  time  the  men  worked,  how  much  should  each  receive  ? 

*  COMPOUND  PROPORTION 

266.    (1)  If  20  men  can  dig  60  yd.  of  earth  in  4  da.,  how 
many  yards  can  30  men  dig  in  9  da.  ? 

Men  Yd.  Da. 

20  60  4 

30  £  9 

Multiply  60  yd.  by  f-g.     v  30  men  can  dig  |§  as  many  yd.  as  20  men. 
Multiply  the  result  by  f .     v  in  9  da.  30  men  can  dig  f  as  much  as  in  4  da. 

yd. 

.-.  x  =  B£  x  |§  x  |  =  202|  yd. 


278  ARITHMETIC 

(2)  If  120  bu.  of  oats  last  14  horses  56  da.,  in  how  many 
da.  will  6  horses  consume  90  bu.  ? 

Bu.  Horses  Da. 

120  14  56 

90  6  x 

Multiply  56  da.  by  &-$.     v  90  bu.  will  last  ^/s  as  many  da.  as  120  bu. 
Multiply  the  result  by  -XF4-.     v  90  bu.  will  last  6  horses  -1/  as  long  as  14 

horses. 

da. 

.-.  x  =  -5T6-  x  T92°o  x  -1/-  -  98  da. 


267.  In  each  of  the  above  two  solutions,  in  order  that  the 
ratio  may  easily  be  seen,  the  items  in  the  question  have  been 
written  in  horizontal  lines.     In  number  (1)  we  are  required 
to  find  the  number  of  yards,  and  the  problem  is  to  determine 
the  ratio  resulting  from  each  comparison,  and  how  it  affects 
the  number  of  yards. 

In  number  (2)  we  are  required  to  find  the  number  of  days, 
and  the  problem  is  to  determine  the  ratios,  and  how  they 
affect  the  number  of  days. 

268.  Both  problems  may,  if  preferred,  be  solved  by  the 
unitary  method,  thus  : 

,,,  20  men  in  4  da.  dig  60  yd. 

1  man  in  4  da.  digs  §§  yd. 

(\f\ 

1  man  in  1  da.  digs  -  yd. 

20  x  4J 

30  men  in  1  da.  dig  6Q  x  30  yd. 

20  x  4 

.-.  30  men  in  9  da.  dig  60  x  3Q  x  9  =  2021  yd. 

20  x  4 

14  horses  eat  120  bu.  in  56  da. 

1  horse  eats  1  bu.  in  -  --  da. 

120 

<\(\  v  14  v  QO 

.  •.  6  horses  eat  90  bu.  in  °    x        x       =  98  da. 

120  x  6 


RATIO   AND   PROPORTION  279 

To  prove  the  answer  correct,  substitute  the  answer  in  place  of  x  in  the 
horizontal  line  and  omit  one  of  the  quantities,  frame  the  question  and  then 

solve. 

Bu.  Horses  Da. 

120  14  56 

90  x  98 

If  120  bu.  of  oats  last  14  horses  for  56  da.,  how  many 
horses  will  90  bu.  last  98  da.  ? 

On  solving,  x  will  be  found  equal  to  6,  which  proves  the  former  solution 
correct.  How  many  questions  can  be  made  from  the  numbers  in  the  two 
lines,  including  the  original  one  ? 

Solve  the  following  questions.  State  one  or  more  questions 
in  proof  for  each  problem,  and  prove  your  answers  correct. 


Exercise  170 

1 .  If  7  horses  are  kept  20  da.  for  $  14,  how  many  will  be  kept 

7  da.  for  $  28  ? 

2.  If  3  men  earn  $  75  in  20  da.,  how  many  men  will  earn 
$  78.75  in  9  da.  at  the  same  rate  ? 

3.  If  16  horses  eat  96  bu.  of  corn  in  42  da.,  in  how  many  days 
will  7  horses  eat  66  bu.  ? 

4.  If  16  horses  can  plough  1280  A.  in  8  da.,  how  many  A.  will 
12  horses  plough  in  5  da.  ? 

5.  If  20  men  can  perform  a  piece  of  work  in  12  da.,  find  the 
number  of  men  who  could  perform  another  piece  of  work  3  times 
as  great  in  ^  of  the  time. 

6.  If  252  men  can  dig  a  trench  210  yd.  long,  3  wide,  and  2 
deep,  in  5  da.  of  11  hr.  each,  in  how  many  days  of  9  hr.  each 
will  22  men  dig  a  trench  of  420  yd.  long,  5  wide,  and  3  deep  ? 

7.  If  10  men  can  reap  a  field  of  7-J-  A.  in  3  da.  of  12  hr.  each, 
how  long  will  it  take  8  men  to  reap  9  A.,  working  16  hr.  a  day  ? 


280  ARITHMETIC 

8.  If  25  men  can  do  a  piece  of  work  in  24  da.,  working  8  hr. 
a  day,  how  many  hours  a  day  would  30  men  have  to  work  in  order 
to  do  the  same  piece  of  work  in  16  da.  ? 

9.  A  town  which  is  defended  by  1200  men,  with  provisions 
enough  to  sustain  them  42  da.,  supposing  each  man  to  receive 
18   oz.  a  day,  obtains  an  increase  of  200  men  to  its  garrison. 
What  must  now  be  the  allowance  to  each  man,  in  order  that  the 
provisions  may  serve  the  whole  garrison  for  54  da.  ? 

10.  If  560  flagstones,  each  11  ft.  square,  will  pave  a  court- 
yard, how  many  will  be  required  for  a  yard  twice  the  size,  each 
flagstone  being  14  in.  by  9  in.  ? 

11.  If  20  men  in  3  wk.  earn  $900,  in  what  time  will  12  men 
earn  $  1500  ? 

12.  If  T9T  of  a  meadow  be  mown  by  12  men  in  6  da.,  find  in 
what  time  the  remainder  could  be  mown  by  10  men. 

13.  If  36  men,  working  16  da.,  can  dig  a  trench  72  yd.  long, 
18  yd.  wide,  and  12  yd.  deep,  how  many  men  can  dig  a  trench 
64  yd.  long,  27  yd.  wide,  and  18  yd.  deep  in  24  da.  ? 

14.  If  25  men  build  a  wall  15  ft.  high,  2  ft.  thick,  and  50  ft. 
long,  in  12  da.  of  9  hr.  each,  how  many  hours  per  day  must  40 
men  work  to  build  a  wall  60  ft.  long,  3  ft.  thick,  and  20  ft.  high 
in  25  da.  ? 

15.  Twenty  men  can  do  a  piece  of  work  in  12  da.     Find  how 
many  men  will  do  half  as  much  again  in  one-fifth  part  of  the 
time,  supposing  that  they  work  the  same  number  of  hours  in  the 
day,  and  that  2  of  the  second  set  can  do  as  much  work  in  an 
hour  as  3  of  the  first  set. 

16.  If  12  men  do  a  piece  of  work  in  21  da.,  in  what  time  will 
10  men  do  a  piece  of  work  If  as  great,  if  3  of  the  first  set  do  as 
much  in  an  hour  as  4  of  the  second  set  ? 

17.  A  miller  has  a  bin  8  ft.  long,  4±-  ft.  wide,  and  2J  ft.  deep, 
holding  75  bu.     How  deep  must  he  make  another  bin  which  is  to 
be  18  ft.  long  and  3|-  ft.  wide,  so  that  its  capacity  may  be  450  bu,? 


RATIO    AND   PROPORTION  281 

18.  What  is  the  weight  of  a  block  of  stone  12  ft.  6  in.  long, 
6  ft.  6  in.  broad,  and  8  ft.  3  in.  deep,  when  a  block  of  the  same 
stone  5  ft.  long,  3  ft.  9  in.  broad,  and  2  ft.  6  in.  deep,  weighs 
7500  Ib.  ? 

PROPORTIONAL  PARTS 

269.    (1)  Divide  $720  into  parts  proportional  to  4,  5,  and  6. 

The  total  number  of  parts  =  4  +  5  +  6  =  1 5. 

.-.  the  first  part  =  T\  of  $  720  =  $  192, 
the  second  part  =  T55  of  $  720  =  $  240, 
the  third  part  =  T65  of  $  720  =  $  288. 

(2)  Divide  316  Ib.  into  parts  proportional  to  J,  |>  and  |. 

Multiplying  i,  i,  and  1  by  their  L.  C.  M.  120,  we  have  the  parts  propor- 
tional to  40,  24,  and  15. 

The  total  number  of  parts  =  40  +  24  +  15  =  79. 

.-.  the  parts  are  respectively  f |,  ff,  and  i-f  of  316  Ib.  =  160,  96,  and  60  Ib. 

PROOF.  —  Dividing  160,  96,  and  60  by  480,  the  denominator  which  reduces 
160  to  |,  we  have  *,  i,  |,  which  proves  the  results  found  to  be  correct. 

Exercise  171 

1.  Divide  1331  into  parts  proportional  to  2,  4,  5. 

2.  Divide  $73.50  into  parts  proportional  to  1,  -|,  J. 

3.  Divide  19  T.  1104  Ib.  into  parts  proportional  to  |,  J-,  \. 

4.  Divide  $  1064  into  parts  proportional  to  2,  2J,  2|. 

5.  Divide  180  Ib.  into  parts  proportional  to  3.3,  .7,  .5. 

6.  Divide  $4500  between  two  persons  in  proportion  to  their 
ages,  which  are  21  and  24  yr. 

7.  Two  men  receive  $15  for  doing  a  certain  piece  of  work. 
Now  one  man  had  worked  only  3  da.,  while  the  other  had  worked 
5  da.  on  the  job.     If  the  money  is  to  be  divided  in  proportion 
to  the  lengths  of  time  the  men  worked,  how  much  should  each 
receive  ? 


282  ARITHMETIC 

8.  Divide  4472  into  parts  which  shall  be  to  each  other  in  the 
ratio  of  3,  5,  7,  11. 

9.  Divide  $84.42  into  two  parts  which  shall  be  to  each  other 
as  5 : 16. 

10.  A  company  of  militia  consisting  of  72  men  is  to  be  raised 
from  3  towns  which  contain  respectively  1500,  7000,  and  9500 
men.     How  many  must  each  town  provide  ? 

11.  Sugar  is  composed  of  49.856  parts  oxygen,  43.265  carbon, 
and  6.879  hydrogen.     How  many  Ib.  of  each  are  there  in  1300  Ib. 
of  sugar  ? 

12.  Gunpowder  is  composed  of  nitre,  charcoal,  and  sulphur  in 
the  proportion  of  33,  7,  and  5. 

(1)  How  many  Ib.  of  sulphur  are  there  in  180  Ib.  of  powder  ? 

(2)  How  many  Ib.  of   powder   can   be  made  with  30   Ib.   of 
sulphur  ? 

(3)  How  much  nitre  and  sulphur  must  be  mixed  with  112  Ib. 
of  charcoal  to  form  gunpowder  ? 

13.  A  man  divides  $3300  amongst  his  three  sons,  whose  ages 
are  16,  19,  and  25  yr.,  in  sums  proportional  to  their  ages;  2  yr. 
afterwards  he  similarly  divides  an  equal  sum,  and  again  after  3  yr. 
more.     How  much  does  each  receive  in  all  ? 

14.  Two  persons   travelling   together  agree   to  pay  expenses 
in  the  ratio  of  7  to  5.     The  first  (who  contributes  the  greater 
sum)  pays  on  the  whole  $103.40,  the  second  $63.40.     What 
must  one  pay  the  other  to  settle  their  expenses  according  to 
agreement  ? 

15.  Divide  $480  among  A,  B,  C,  and  D,  so  that  B  may  receive 
as  much  as  A ;  C  as  much  as  A  and  B  together ;  and  D  as  much 
as  A,  B,  and  C  together. 


RATIO   AND   PROPORTION  283 

PARTNERSHIP 

270.  In  Simple  Partnership  the  capital  of  each  partner  is 
supposed  to  be  invested  for  the  same  time. 

In  Compound  Partnership  the  time  is  taken  into  account  as 
well  as  the  capital  in  determining  the  gain  or  loss  of  each 
partner. 

271.  A,  B,  and  C  engage  in  business.    A  furnishes  $7500, 
B  15000,  and  C  $4500.     If  they  gain  $2380,  what  is  each 
one's  share  ? 

Dividing  their  capitals  by  $  500,  we  have  their  capitals,  and  therefore  their 
gains  proportional  to  15,  10,  and  9. 

The  total  number  of  parts  —  15  +  10  +  9  =  34. 

.-.  their  respective  gains  are  ££,  if,  and  /T  of  $2380  =  $1050,  $700,  and 
$030. 

Exercise  172 

1.  Two  merchants,  A  and  B,  form  a  joint  capital.     A  puts  in 
$  1200  and  B  $  1800.     They  gain  $  400.     How  ought  the  gain  to 
be  divided  between  them  ? 

2.  A  bankrupt  owes  three  creditors,  A,  B,  and  C,  $175,  $210, 
and  $265  respectively;   his  property  is  worth  $422.50.     What 
ought  each  to  receive  ? 

3.  A,  B,  and  C  entered  into  partnership.      A  puts  in  $6000, 
B  $4000,  and  C  $2000.     They  gained  $2250.     What  is  each 
one's  share  of  the  gain  ? 

4.  Two  men  purchase  a  house  for  $3600,  the  first  contributing 
$1600  and  the  second  $2000.     If  it  rents  so  as  to  pay  12%  on 
its  value,  what  share  of  the  rent  should  each  receive  ? 

5.  Two  persons  have  gained  in  trade  $3456;  the  one  put  in 
$  10,560  and  the  other  $  8640.     What  is  each  person's  share  of 
the  profits  ? 


284  ARITHMETIC 

6.  R.  Stuart  and  G.  Armstrong  enter  into  partnership.    Stuart 
contributes  $  4500  to  the  partnership  and  Armstrong  contributes 
$  7500.     Their  net  gain  at  the  end  of  the  year  is  $ 1750.     How 
much  of  the  sum  should  each  partner  receive  ? 

7.  Three   partners    invest   respectively    $7800,    $5750,    and 
$  9450  in  business.     At  the  end  of  the  first  year  they  find  their 
net  gain  to  be  $3156.     What  is  the  amount  of  each  partner's 
share  of  this  gain  ? 

8.  A,  B,  and  C  form  a  partnership  with  a  capital  of  $20,000. 
A  contributes  $  5000,  B  $  7000,  and  C  the  remainder.     They  gain 
20%  of  the  total  capital.     Find  each  man's  share  of  the  profits. 

9.  T.  Allan  and  E.  Jamieson  engage  in  business  with  a  joint 
capital  of  $  19,200,  and  agree  to  share  gains  and  losses  in  propor- 
tion to  their  investments.     At  the  end  of  a  year  Allan  receives  a 
dividend  of  $1100,  and  Jamieson  a  dividend  of  $1300.     What 
was  the  amount  of  the  investment  of  each  ? 

10.  D.  Rowan,  F.  Galbraith,  and  J.  Munro  enter  into  partner- 
ship.    They  gain  $7500,  of  which  Rowan  receives  $2100,  Gal- 
braith $3100,  and  Munro  the  balance.     How  much  did  Rowan 
and   Galbraith   respectively   invest   if   the   amount   of   Munro's 
investment  was  $18,000? 

11.  A,  B,  and  C  pay  $37.80  as  rent  for  a  pasture.     A  puts  in 
5  horses,  B  12  cows,  and  C  60  sheep.     If  1  horse  eats  as  much  as 
2  cows,  and  1  cow  as  much  as  3  sheep,  what  rent  should  each 
pay? 

272.  (1)  A,  B,  and  C  enter  into  partnership.  A  puts  in 
$700  for  12  mo.,  B  $500  for  9  mo.,  and  C  1 600  for  8  mo. 
Divide  a  profit  of  $2065  equitably  among  them. 

The  gain  on  $  700  for  12  mo.  =  the  gain  on  $  8400  for  1  mo. 
The  gain  on  $  500  for    9  mo.  —  the  gain  on  $4500  for  1  mo. 
The  gain  on  $  600  for    8  mo.  =  the  gain  on  $  4800  for  1  mo. 
/.  the  proportional  parts  representing  the  gains  are  84,  45,  and  48,  or  28, 
15,  and  16.  28  +  15  -f  16  =  59. 

.-.  the  respective  gains  are  ?f,  £»,  and  |f  of  $  2065  =  $980,  $  525,  and  $  560. 


RATIO    AND   PROPORTION  285 

(2)  A  commenced  business  with  $4000  stock  ;  3  mo.  after, 
he  took  in  B  with  a  capital  of  $2000;  and  4  mo.  after  B 
became  a  partner,  he  took  in  C  with  a  capital  of  $600;  at 
the  end  of  the  year  the  firm  had  gained  $3450.  Find  the 
share  of  each. 

A's  capital  =  §4000  for  12  mo.  -  $48,000  for  1  mo. 
B's  capital  =  $2000  for  9  mo.  =  $  18,000  for  1  mo. 
C's  capital  =  $  600  for  5  mo.  =  $  3000  for  1  mo. 

/.  the  respective  gains  are  proportional  to  48,  18,  and  3  ;  i.e.  16,  6,  and  1. 

16  +  6  +  1  =  23. 

/.  the  respective  shares  are  if,  56F,  and  ^  of  $3450  =  $2400,  $900,  and 
$150. 

Exercise  173 

1.  D  and  E  enter  into  partnership ;  D  puts  in  $480  for  3  mo., 
and  E  $  900  for  4  mo.     They  gain  $  840.     What  is  each  man's 
share  in  the  gain  ? 

2.  A,  B,  and  C  entered  into  partnership ;  A  put  in  $  1200  for 

8  mo.,  B  $  800  for  10  mo.,  and  C  $  400  for  12  mo.     They  gained 
$  3920.     What  was  each  man's  share  of  the  gain.  ? 

3.  A,  B,  and  C  are   partners ;    A  puts  in  $  5000  for  6  mo., 
B  $6000   for   8    mo.,   and    C    $900    for    11    mo.     The  profit  is 
$  5575.50.     What  is  the  share  of  each  ? 

4.  Three  graziers  hire  a  pasture  for  their  common  use,  for 
which  they  pay  $  318.     One  puts  in  20  oxen  for  6  mo.,  another 
24  oxen  for  8  mo.,  and  the  third  28  oxen  for  4  mo.     How  much 
of  the  rent  should  each  pay  ? 

5.  A  and  B  enter  into  partnership  ;  A  contributes  $  15,000  for 

9  mo.,  and  B  $  12,000  for  6  mo.     They  gain  $  5750.     Find  each 
man's  share  of  the  gain. 

6.  A,  B,  and  C  rent  a  field  for  $  56.50 ;  A  puts  in  70  cattle 
for  6  mo.,  B  40  for  9  mo.,  and  C  50  for  7  mo.     What  ought  each 
to  pay  ? 


286  ARITHMETIC 

7.  Three  merchants  enter  into  partnership;  the  first  invests 
$  1855  for  7  mo.,  the  second  invests  $  887.50  for  10  mo.,  and  the 
third  invests  $  770  for  11  mo.,  and  they  gain  $  434.    What  should 
be  each  partner's  share  of  the  gain  ? 

8.  L,  M,  and  N  entered  into  partnership  and  invested  respec- 
tively $  19,200,  f  22,500,  and  $  28,300.     At  the  end  of  5  mo.  L 
invested  $  3800  additional,  M  $  2500,  and  N  $  3700.     At  the  end 
of  a  year  the  net  gain  of  the  firm  was  found  to  be  $  7850.     What 
was  each  partner's  share  of  this  ? 

9.  A  and  B  enter  into  partnership ;  A  puts  in  $  400  at  first, 
and  $  500  at  the  end  of  2  mo. ;   B  puts  in  $  300  at  first,  and 
$  600  at  the  end  of  3  mo.     The  profit  at  the  end  of  the  year 
is  $  470.     How  should  this  be  divided  ? 

10.  A  and  B  engage  in  trade ;  A  invests  $  6000,  and  at  the 
end  of  5  mo.  withdraws  $  2000 ;  B  puts  into  the  business  $  4000, 
and  at  the  end  of  7  mo.  $  6000  more.     Divide  a  gain  of  $  6800  at 
the  end  of  the  year. 

11.  A,  B,  and  C  form  a  partnership  with  a  joint  stock  of 
$15,600;   A's  stock  continues  in  trade  6  mo.,  B's  8  mo.,  and 
C's  12  mo.    A's  gain  is  $  1200,  B's  $  2400,  and  C's  $  1800.    What 
stock  did  each  put  in  ? 

12.  Two  men  complete  in  a  fortnight  a  piece  of  work  for  which 
they  are  paid  $46.75;  one  of  them  works  alternately  9  hr.  and 
8  hr.  a  day,  the  other  works  81  hr.  for  5  da.  in  the  week,  and 
does  nothing  on  the  remaining   day.      What  part  of   the  sum 
should  each  receive  ? 

13.  A  and  B  are  partners;  A's  capital  is  to  B's  as  4  to  9.     At 
the  end  of  4  mo.  A  withdraws  J  of  his  capital,  and  B  |  of  his. 
At  the  end  of  the  year  their  whole  gain  is  $  4600.     How  much 
belongs  to  each  ? 

14.  A,  B,  and  C  rent  a  pasture  for  $  92 ;  A  puts  in  6  horses  for 
8  wk.,  B  12  oxen  for  10  wk.,  C  50  cows  for  12  wk.     If  5  cows  are 


RATIO   AND   PROPORTION  287 

reckoned  equivalent  to  3  oxen,  and  4  oxen  to  3  horses,  what 
should  each  pay  ? 

15.  Three  men,  working  respectively  8,  9,  and  10  hr.  a  day, 
receive  the  same  daily  wages.    After  working  thus  for  3  da.,  each 
works  1  hr.  a  day  longer,  and  the  work  is  finished  in  3  da.  more. 
If  $  114  is  paid  for  the  work,  how  much  should  each  man  receive  ? 

16.  Three  workmen,  A,  B,  and  C,  did  a  certain  piece  of  work 
and  were  paid  daily  wages  according  to  their  several  degrees  of 
skill.    A's  efficiency  was  to  B's  as  4  to  3,  and  C's  to  B's  as  5  to  6 ; 
A  worked  5  da.,  B  6  da.,  and  C  8  da.     The  whole  amount  paid 
for  the  work  was  $  36.25.     Find  each  man's  daily  wages. 


CHAPTER   XVI 


POWERS  AND  BOOTS 

SQUARE  ROOT* 
273.    (1)  Find  the  square  root  of  17.3056. 

17  .30'56  |  4.16 
16 


81 


826 


130 
81 


4956 
4956 


To  prove  4.16  the  right  answer,  square  4.16  and  the  result  will  be  found 
to  be  17.3056. 

(2)  Extract  the  square  root  of  35  to  three  decimal  places. 

35  |  5.916 

25 


109 


1181 


11826 


1000 
981 


1900 
1181 


71900 
70956 


(3)  Extract  the  square  roots  of  ||-,  f  f ,  f . 

3      V25      5 


*  See  Chapter  VI II. 


V49      7 

288 


POWERS   AND   ROOTS  289 


/35_  V35_5.916 

>/49~    _/77^~ 


V    =  V^625  =  .7905. 


Which  denominator  is  not  a  perfect  square  ?    Why  reduce  f  to  a  decimal 
before  extracting  the  square  root  ? 

Exercise  174 

Find  the  square  root  of  : 

1.  40.96;  65.61;  2.1025. 

2.  167.9616;  28.8369;  57648.01. 

3.  .042849;  .00139876;  .00203401. 

4.  5774409;  5.774409. 

5.  10.3041;  2321.3124;  .0050367409. 


6.  V2;  V20;  V.4;  VlOOO  to  four  decimal  places. 

7  144  .     324  .    fil 

'  '  ~2~S~5  J    T6  1  '    u¥' 

ft  201  •1-56.1.     2209 

O.  ^V^,     1-Yg-gj    -3)     98Ul' 


3.5.      7 
-      8"  5    "9"'     1  1' 


*  CUBE  ROOT 

274.  The  product  of  3  x  3  x  3  is  27 ;  of  5  x  5  x  5  is  125. 
The  cubes  whose  sides  measure  3  and  5  units  of  length  con- 
tain 27  and  125  units  of  volume.     We  say  that  27  is  the 
cube  of  3  and  that  125  is  the  cube  of  5;  that  3  is  the  cube 
root  of  27,  and  that  5  is  the  cube  root  of  125.     The  cube^)f 
5  is  written  53,  and  the  cube  root  of  5  is  indicated  thus :  V  5. 

53  is  also  called  the  third  power  of  5. 

275.  The  cubes  of 

1,  2,  3,    4,     5,      6,      7,      8,      9,      10, 
are  1,  8,  27,  64,  125,  216,  343,  512,  729,  1000. 

u 


290  ARITHMETIC 

276.  The  cube  roots  of 

1,  8,  27,  64,  125,  216,  343,  512,  729,  1000, 
are  1,  2,  3,    4,     5,      6,      7,      8,      9,      10. 

277.  These  two  paragraphs  should  be    mastered   by  the 
pupils  as  the  corresponding  paragraphs  in  square  root. 

278.  The  product  4  x  4  x  4  is  written  43. 

The  product  4.6  x  4.6  x  4.6  is  written  (4.6)3. 
The  product  f  x  |  x  f  is  written  (|)3. 
The  cube  root  of  4  is  written  VI. 
The  cube  root  of  ,4  is  written 

o 

The  cube  root  of  f  is  written 

Exercise  175 

Write  the  following  products  as  powers : 

1.  2  x  2  x  2.  7.  4  x  4  x  4, 

2.  3  x  3.  8.  |  x  |  X  £ 

3.  5  x  5  x  5.  9.  2.3  x  2.3  x  2.3. 

4.  5  x  5  x  5  x  5.  10.  3.12  x  3.12  x  3.12. 

5.  5  x  5  x  5  x  5  x  5.  11.  .12  x  .12  x  .12. 

6.  5x5x5x5x5x5.  12.  .1  x  .1  x  .1. 

13.   .02  x  .02  x  .02. 

Write  the  following  powers  as  products  and  find  their  values : 
14.    43.  15.    123.  16.    2.53.  17.    (f)3.  18.    (i)3. 

19.    .023.  20.    .I3. 


POWERS   AND   ROOTS  291 

Prove  the  following  statements : 


21.    \26  =  6.  26. 


T- 


22.  S/15625  =  25.  27.    V^8T  =  -f. 

23.  A/15.625  =  2.5.  28.    -3/T^  -  5 

29. 
30. 


Exercise  176 

1.  Find  the  length  of  one  edge  of  a  cube  containing  512  cu. 
in.     Find  the  length  of  all  its  edges.     Find  the  area  of  one  of  its 
faces.     Of  all  its  faces. 

2.  Find  the  area  of  one  face  of  a  cube  containing  729  cu.  in. 

3.  Find  the  number  of  units  of  length  in  a  cube  containing 
343  units  of  volume.     Find  the  number  of  units  of  area  in  one 
face. 

4.  Find  the  edge  of  a  cube  one  of  whose  faces  contains  144 
sq.  in.     Find  its  volume. 

5.  Find  the  volume  of  a  cube  one  of  whose  faces  contains 
225  sq.  in. 

6.  What  is  the  edge  of  a  cube  whose  volume  is  8  units  of  vol- 
ume ?     27  units  of  volume  ? 

7.  The  ratio  of  the  volumes  of  two  cubes  is  ^.     What  is  the 

£t    I 

ratio  of  their  edges  ? 

8.  The  ratio  of  the  volumes  of  two  cubes  is  64 : 125.    What  is 
the  ratio  of  their  edges  ? 

9.  The  ratio  of  the  edges  of  two  cubes  is  -f-.     What  is  the 
ratio  of  their  volumes  ? 

10.    The  edges  of  two  cubes  are  as  7  :  9.     What  is  the  ratio  of 
their  volumes  ? 


292  ARITHMETIC 

Exercise  177 

1.  Find  the  cubes  of  14,  25,  36,  54,  75,  and  99. 

2.  From  the  results  in  §  275,  state  how  many  digits  there  are 
in  the  cube  of  a  number  of  1  digit. 

3.  From    the   results   in   question   1,   state  how  many  digits 
there  are  in  the  cube  of  a  number  of  2  digits. 

4.  In  long  division  how  many  figures   form  a  group  ?      In 
square  root  ?     In  cube  root  ? 

5.  How  many  digits  are  there  in  the  cube  root  of  512?     64? 
8  ?     What  are  they  ? 

6.  Divide  389,017  into  groups  of  figures ;  29,791 ;  3375.    How 
many  figures  are  there  in  the  cube  root  of  each  of  these  numbers? 

7.  Cube  73,  31,  and  15. 

8.  Judging  from  the  results  given  in  §  276,  state  the  number 
of  digits  in  the  cube  root  of  a  number  containing  1,  2,  or  3  digits. 

9.  Write  the  cubes  of  10,  20,  30,  40,  50,  60,  70,  80,  and  90. 

10.  State  how  many  digits  there  are  in  the  cube  root  of  a  num- 
ber containing  4,  5,  or  6  digits. 

11.  What  is  the  first  digit  in  the  cube  root  of  2744  ?     39,304? 

279.    To  find  the  cube  root  of  a  number,  we  shall  first  see 
how  the  cube  of  a  number  is  found. 

Since  54  =  50  +  4,  we  can  cube  54  thus : 

50  +  4 

50  +  4 


(4  x  50)  +  42 
5Q2  +  (4  x  50) 
502  +  2(4  x  50) +42 

50  +  4 

(4  x  502)+  2(42  x  50) +  43 
503  +  2(4  x  502)  +    (42x50) 
503  +  3(4  x  50'2)  +  3(42  x  50)  +  43 

Since   4   divides   each   of  the   last   three   terms,  we   can  put  this  result 
=  503  +  4(3  x  502  +  3  x  4  x  50  +  42}. 


POWERS   AND    ROOTS 


293 


We  now  wish  to  recover  from  such  a  number  as  157,464  its  cube  root. 
Plainly  the  tens'  digit  of  the  root  is  5,  i.e.  the  first  part  of  the  root  is  50. 

50     157'464  [  50  +  4 

125000 

3  x  502  =  7500      32464 
3  x  4  x  50  =    600 
42=      16 


8116 


32464 


To  find  the  second  term,  note  that  in  the  expression 

4  {3  x  502  +  3  x  4  x  50  +  42} 

the  number  4  is  the  second  digit  in  the  number  54  that  was  cubed  ;  hence  in 
the  work  of  taking  the  cube  root,  the  other  factor  {3  x  50-+  3  x  4  x  50  +  42 } 
will  be  the  real  divisor  and  3  x  502  the  trial  divisor. 

Therefore,  squaring  50  and  multiplying  by  3,  we  have  7500.  Dividing  7500 
into  32,464,  we  find  the  quotient  to  be  4  ;  completing  the  divisor  by  adding 
3  x  4  x  50  or  600,  and  42  or  16,  we  find  the  divisor  to  be  7500  +  600  +  16 
or  8116.  Multiplying  this  by  4,  we  have  32,464.  Hence  we  conclude  that 
the  cube  root  of  157,464  is  54.  To  prove  the  result  correct,  cube  54. 

280.   The  work  of  extracting  the  cube  root  may  be  shortened 

thus : 

157'464[54 

125 


300  x  52  =  7500 

30  x  5  x  4   =    600 

42=    J^ 

8116 


32464 


32464 


Find  the  cube  root  of  926,859,375. 


300  x  92  =  24300 
30  x  9  x  7  =  1890 

72= 49 

26239 

300  x  972  =  2822700 
30  x  97  x  5  =   14550 

52  = 25 

2837275 


926'859'375|975 
729 


197859 


183673 


14186376 


14186375 


294 


ARITHMETIC 


281.  The  cube  of  3.19  is  equal  to  32.461759.  From  this 
it  is  evident  that  corresponding  to  the  two  figures  in  the 
decimal  part  of  the  number,  viz.  19,  we  have  two  groups  of 
three  figures,  viz.  461  and  759,  in  the  decimal  part  of  the 
cube.  Hence  in  pointing  off,  begin  at  the  decimal  point  and 
mark  the  number  off  into  periods  of  three  figures  each  to  the 
right  of  the  decimal  and  then  again  to  the  left. 


(1)  Find  the  cube  root  of  95.443993. 

95.443'993 
^64 

300  x  42  =  4800 
30  x  4  x  5  =  600 
52=  25 


5425 

300  x  452  =  607500 
30  x  45  x  7  =   9450 

72  = 49 

616999 


4.67 


31443 


27125 


4318993 


4318993 


(2)  Extract  the  cube  root  of  16. 


300  x  22  =  1200 

30  x  2  x  5  =  300 

52=   25 

1525 

300  x  252  =  187500 

30  x  25  x  1  =   750 

12  = 1 

188251 

300  x  2512  _  18900300 
30  x  251  x  9  =   667700 

92= 81 

19568081 


16 

8 


2.519 


8000 


7625 


375000 


188251 


186749000 


176112729 


(3)  To  extract  the  cube  root  of  such  a  number  as  843.7295, 
add  ciphers  thus,  843.  729'500,  and  extract  the  cube  root. 


POWERS   AND   ROOTS  295 

(4)  Extract  the  cube  root  of 


'343  "",3/343  "7 
(5)  Extract  the  cube  root  of  ^3-. 

=  §^i»  =  .  359. 


'343"  ^343         7 

(6)  Extract  the  cube  root  of  ^~f. 

if  =  .64  or  .640. 
^."640  =  .861. 
...  v/Jf  =  .861. 

Which  of  the  three  denominators  is  not  a  perfect  cube  ?    When  should 
a  fraction  be  reduced  to  a  decimal  before  extracting  its  cube  root  ?     Why  ? 

Exercise  178 

Find  the  cube  root  of : 

1.   29,791.  2.   54,872.  3.   110,592. 

4.   804,357.  5.   941,192. 

6.  Compare  the  processes  of  long  division,  of  extracting  the 
square   root,  and   the  cube  root  of  a   number.     Note   in  what 
respect  they  are  similar  and  in  what  respect  they  are  different. 

7.  2,406,104.  13.    .001906624. 

8.  69,426,531.-  14.   3,  .3,  .03,  .003,  .0003. 

9.  8,365,427.  15.   fr,  iff,  T¥rVr 

10.  389.017.  16.    Iff,  iff 

11.  32.461759.  17.    f,  ^,  f 

12.  .000912673.  18.   3f,  405^,  7f 

19.  A  cubical  block  of  stone  contains  50,653  cu.  ft.     What 
is  the  area  of  its  side  ? 

20.  A  cube  contains  56  cu.  ft.  568  cu.  in.     Find  its  edge. 


296  ARITHMETIC 

21.  One  gal.  contains  231  cu.  in.     Find  the  edge  of  a  cube 
equal  to  it. 

22.  Find  the  length  of  the  inside  edge  of  a  cubical  vessel  which 
will  just  hold  10  gal. 

23.  Three  cubes  of  lead,  measuring  respectively  |,  f,  and  |  of 
an  in.  on  the  edge,  were  melted  together  and  cast  into  a  single 
cube.     Find  the  length  of  the  edge  of  the  cube  thus  formed, 
neglecting  loss  of  lead  in  melting  and  casting. 

24.  Four  cubes  of  lead,  measuring  respectively  6,  7,  8,  and  9  in. 
on  the  edge,  were  melted  together  and  cast  into  a  single  cube. 
Find  the  length  of  the  edge  of  the  cube  thus  formed,  if  4%  of 
the  lead  was  lost  in  melting  and  casting. 

25.  Find  the  volume  of  a  cube,  the  area  of  whose  surface  is 
100.86  sq.  in. 

26.  A  cube  measures  5  in.  on  the  edge.     A  second  cube  is 
three  times  the  volume  of   the  first.     By  how  much    does  the 
length  of  an  edge  of  the  second  cube  exceed  that  of  an  edge  of 
the  first  cube  ? 

27.  By  raising  the  temperature  of  a  cube  of  iron,  the  length 
of  each  of  its  edges  was  increased  by  5%.     Find  correct  to  four 
decimals  the  ratio  of  increase  in  the  volume  of  the  cube. 

28.  Each  edge  of  a  cube  is  diminished  by  TL  of  its  length. 
By  what  fraction  of  itself  is  the  volume  diminished  ?     By  what 
fraction  of  itself  is  the  area  of  the  surface  diminished  ? 


CHAPTER   XVII 

MENSUKATION 

282.   The  rectangle  has  been  treated  of  in  preceding  para- 
graphs. 

Exercise  179 

1.  The  units  of  lengths  being  1  in.,  2  in.,  3  in.,  4  in.,  5  in.,  6  in., 
what  are  the  units  of  area?     Make  mental  pictures  of  the  units 
of  area. 

2.  If  the  length  and  breadth  of  a  rectangle  are  respectively 
7  and  5  times  the  unit  of  length,  how  many  times  the  unit  of  area 
is  the  area  of  the  rectangle  ?     If  the  unit  of  length  is  6  in.,  what 
is  the  area  of  the  rectangle  ? 

3.  If  the  ratios  of  the  length  and  breadth  of  a  rectangle  to  the 
unit  of  length  are  respectively  5  and  3,  what  is  the  ratio  of  the 
area  of  the  rectangle  to  the  unit  of  area  ? 

4.  How  do  you  find  the  number  of  units  of  area  in  a  rectangle  ? 

5.  The  measure  of  the  area  of  a  rectangle  is  48,  the  area  being 
432  sq.  in.     What  is  the  unit  of  area  ?     Of  length  ? 

6.  Draw  a  rectangle  2  in.  by  3  in.  in  one  corner  of  a  rectangle 
6  in.  by  9  in.     Are  the  figures  similar  in  shape  ?     Find  the  area 
of  each  rectangle.     What  is  the  ratio  of  the  two  breadths  ?     Of 
the  two  lengths  ?     Of  the  two  areas  ?      What  is  the  relation 
between  the  ratio  of  the  areas  and  the  ratio  of  the  lengths  ? 

7.  Draw  a  rectangle  3  in.  by  4  in.  in  one  corner  of  a  rectangle 
6  in.  by  8  in.     Are  the  figures  similar  ?     What  is  the  ratio  of  the 
breadths  ?     Of   the   lengths  ?      What,  then,  is  the  ratio   of  the 
areas  ?      Find  the  area  of  each  rectangle  and   prove  your  last 
result  correct. 

297 


298  ARITHMETIC 

8.  The  sides  of  a  rectangle  are  4  in.  and  6  in.  respectively. 
How  often  will  it  measure  another  rectangle  whose  sides  are 
twice  as  long  ?     3  times  as  long  ?     4  times  ?     5  times  ? 

9.  The  unit  of  area  is  a  rectangle  3  in.  by  5  in.     How  often 
will  it  measure  a  rectangle  each  of  whose  sides  is  6  times  as  long  ? 
7  times  ?     8  times  ?     9  times  ? 

10.  Two  rectangles  are  similar  in  shape,  and  the  second  is  4 
times  as  large  as  the  first.     Compare  the  lengths  of  their  sides. 
Illustrate  by  a  drawing  and  make  a  mental  picture. 

11.  The  areas  of  two  similar  rectangles  are  as  9 : 1.    What  is  the 
ratio  of  their  sides  ? 

12.  The  ratio  of  the  areas  of  two  similar  rectangles  is  9:4. 
What  is  the  ratio  of  their  sides  ? 

13.  Make  a  rule  showing  how  to  find  the  ratio  of  the  sides  of 
two  similar  rectangles  when  you  know  the  ratio  of  their  areas. 

14.  What  are  the  ratios  of  the  areas  of  the  following  pairs  of 
similar  rectangles,  the  ratios  of  the  sides  being  4  to  1 ;   5  to  1 ; 
3  to  4 ;  7  to  2  ?     If  the  area  of  the  smaller  rectangle  is  16  sq.  ft., 
what  is  that  of  the  larger  in  each  case  ? 

15.  What  are  the  ratios  of  the  sides  of  the  following  pairs  of 
similar  rectangles,  the  ratios  of  the  areas  being  64  to  1 ;  121  to  1 ; 
49  to  25 ;  144  to  100  ?     If  the  sides  of  the  smaller  rectangle  are 
20  and  30  in.,  what  are  the  sides  of  the  larger  rectangles  ? 

283.    (1)  Find  to  the  nearest  inch  the  side  of  a  square 

whose  area  is  1  A. 

1  A.  =  6,272,640  sq.  in. 
The  measure  of  the  area  =  6,272,640. 
The  measure  of  the  side  =  2504.5  + . 
/.  the  side  of  the  square,  to  the  nearest  in.  =  2505  in.  =  69  yd.  1  ft.  9  in. 

(2)  The  lengths  of  the  sides  of  a  rectangular  piece  of  land 
are  as  3  to  8  and  its  area  is  60  A.     Find  the  length  of  its  sides. 


MENSURATION  299 

Imagine  the  length  of  the  field  to  be  divided  into  8  equal  parts  and  the 
breadth  into  3,  and  the  field  be  divided  into  24  equal  squares  by  lines  drawn 
through  these  points  of  division  parallel  to  the  sides  of  the  field.     Consider 
one  of  these  squares  the  unit  of  area  and  one  of  its  sides  the  unit  of  length. 
Then  24  x  the  unit  of  area  =  60  A.  or  600  sq.  ch. 

The  unit  of  area  =  25  sq.  ch. 
The  unit  of  length  =  5  ch. 
/.  the  lengths  of  the  sides  =  15  ch.  and  40  ch. 

Exercise  180 

1 .  A  square  field  contains  exactly  8  A.     Determine  the  length 
of  a  side  of  the  field,  correct  to  the  nearest  link. 

2.  The  area  of  a  chess-board  marked  in  8  rows  of  8  squares 
each  is  100  sq.  in.     Find  the  length  of  a  side  of  a  square. 

3.  On  a  certain  map  it  is  found  that  an  area  of  16,000  A.  is 
represented  by  an  area  of  6.25  sq.  in.     Give  the  scale  of  the  map 
in  miles  to  the  inch. 

4.  A  rectangle  measures  18'  by  30'.     Find  the  difference  be- 
tween its  area  and  that  of  a  square  of  equal  perimeter. 

5.  Two  rectangular  fields  are  of  equal  area.   One  field  measures 
15  ch.  by  20  ch. ;   the  other  is  square.     Find  the  length  of  a  side 
of  the  latter  field,  correct  to  the  nearest  link. 

6.  How  many  stalks  of  wheat  could  grow  on  1  A.  of  ground, 
allowing  each  stalk  a  rectangular  space  of  2"  by  3"? 

7.  How  many  pieces  of  turf  3'  6"  by  1'  3"  will  be  required  to 
sod  a  rectangular  lawn  28'  by  60'? 

8.  Sidewalks  12  ft.  wide  are  laid  on  both  sides  of  a  street  440 
yd.  long.     Find  the  cost  of  the  sidewalks  at  $  1.35  per  sq.  yd.  for 
the  pavement  and  75^  per  lineal  yd.    for   curbing ;    deducting 
three  crossings  of  54  ft.  each  on  both  sides  of  the  street. 

9.  The  area  of  a  rectangular  field  is  15  A. ;  the  length  of  the 
field  is  double  the  width.    Find  the  length  of  the  field  in  chains. 

10.    What  length  must  be  cut  off  a  board,  which  is  1\  in.  broad, 
so  that  the  area  may  contain  3  sq.  ft.  ? 


300 


ARITHMETIC 


11.  Find  the  area  of  each  of  the  following  rectangles,  whose 
dimensions  are  :    1.    L  =  36  ft,  B  =  13  ft.     2.    L  =  20  ft.  3  in., 
B  =  20  in.     3.   L  =  8  f t.  9  in.,  B  =  3  ft  8  in. 

12.  Find  the  solid  contents  of  the  following :  1.  L  —  13  ft.  4  in., 
B  =  7  ft.  6  in.,  £T=  3  ft.  10  in.     2.    L  =  20  ft,  73=1  ft.  6  in., 
H=l  ft  2  in. 

13.  Find  the  length  of  the  following  rectangles:    1.    Area  = 
40  sq.  yd.,  B  =  20  ft.     2.  Area  =  6  sq.  ft.,  B  =  9  in. 

14.  Find  the  area  of  the  four  walls  of  the  following  rooms: 
1.  L  =  32  ft,  B  =  18  ft.,  #=  11  ft     2.  i  =  29  ft,  £  =  23^  ft, 
,ff=ili  ft. 

15.  Find  the  cost  of  painting  a  surface :   1.  19  ft.  6  in.  by  83  ft. 
4  in.,  at  40^  a  sq.  ft.     2.  25  ft.  8  in.  long,  and  16  ft.  9  in.  wide,  at 
65^  a  sq.  ft. 

16.  A  gentleman  wishes  to  set  out  a  rectangular  orchard  of 
1260  trees,  so  placed  that  the  number  of  rows  shall  be  to  the 
number  of  trees  in  a  row  as  5  to  7.     Find  the  number  of  rows 
and  also  the  number  of  trees  in  a  row. 

17.  The  sides  of  a  rectangular  field  containing  27  A.  48  sq.  rd. 
are  as  21  to  13.     Find  the  perimeter  of  the  field 

284.    A  Quadrilateral  is  a  plane  figure  having  four  sides. 

A  Parallelogram  is  a  quadrilateral  whose  opposite  sides  are 
parallel. 

A  B 


a 


MENSURATION 


301 


285.  To  find  the  area  of  a  parallelogram : 

Let  perpendiculars  be  drawn  from  C  and  D  perpendicular  to  AB.  Then 
it  is  evident  that  the  triangles  marked  a  are  equal.  Adding  to  each  the 
quadrilateral  marked  c,  it  is  evident  that  the  parallelogram  ABCD  is  equal 
to  the  rectangle  upon  the  base  CD. 

Hence,  to  find  the  measure  of  the  area  of  a  parallelogram, 
multiply  the  measure  of  its  base  by  the  measure  of  its  altitude. 

Draw  any  parallelogram  and  draw  the  perpendiculars  as  in  the  figure. 
Cut  out  the  triangles  and  place  them  on  each  other,  showing  that  they  are 
equal. 

286.  A  Trapezoid  is  a  quadrilateral  two  of  whose  sides  are 
parallel. 

The  parallel  sides  are  called  Bases  and  the  perpendicular 
distance  between  the  two  bases  is  called  the  Altitude. 


D  E  C 

Thus,  in  the  trapezoid,  AB  and  CD  are  the  bases,  and  AE 
the  altitude. 

287.    To  find  the  area  of  a  trapezoid: 

&  B          M 


a 


D      0 


a 


N         C 


302  ARITHMETIC 

Let  ABCD  be  a  trapezoid,  and  let  perpendiculars  be  drawn  through  E 
and  F,  the  middle  points  of  AD  and  SC,  to  AB  and  CD.  Then  it  is  evident 
that  the  triangles  a  and  a'  are  equal  and  also  c  and  c'. 

To  a'  and  c'  and  also  to  a  and  c  add  the  figure  ABFNOE,  and  we  have 
the  trapezoid  equal  to  the  rectangle  LMNO. 

Again, 

EF  =  AB  +  AL  +  BM, 

EF=  CD-DO-NC. 
Adding,  2  EF  =  AB  +  CD,  since 

AL  =  DO  and  BM=  NC, 
i.e.  EF=\(AB  +  CD}. 
.-.  ON  =  \(AB+  CD}. 

Therefore,  to  measure  the  area  of  the  trapezoid,  we  multiply  the  measure 
of  ON,  i.e.  of  \(AB  +  CD),  by  that  of  the  altitude. 

Hence  the  area  of  a  trapezoid  is  found  ly  multiplying  the 
measure  of  one-half  the  sum  of  its  parallel  sides  by  the  measure 
of  its  altitude. 

288.  Find  the  area  of  a  trapezoid  whose  parallel  sides 
are  12'  1"  and  19'  3"  respectively,  the  perpendicular  dis- 
tance between  them  being  8'  5". 

The  sum  of  the  bases  =  12'  7"  +  19'  3"  =  31'  10". 
i  the  sum  of  the  bases  =  15'  11"  =  191". 

The  altitude  =  8' 5"  =  101". 
.-.  the  area  =  101  x  191  sq.  in.  =  19,291  sq.  in.  =  133  sq.  ft.  139  sq.  in. 

Exercise  181 

1.  Fold  a  sheet  of  paper  in  the  form  of  a  trapezoid.     Draw 
the  lines  as  given  in  the  figure  in  §  287,  cut  out  the  triangles 
a  and  c,  and,  by  placing  them   on  a'  and  c'  respectively,  show 
that  a  =  a1  and  c  =  c'. 

2.  In    the   figure  in  question  1,   produce   DC  to   6r,   making 
CG  =  AB.     Measure  along  DG  twice  with  a  line  equal  to  EF, 
thus  showing  that  twice  EF  =  CD  +  AB. 


MENSURATION 


303 


3.  The  length  of  the  base  of  a  parallelogram  is  45  ft. ;  the 
length  of  the  perpendicular  on  the  base  from  the  opposite  side 
is  28  ft.     Find  the  area. 

4.  The  adjacent  sides  of  a  parallelogram  measure  132  ft.  and 
84  ft.  respectively,  and  the  area  of  the  parallelogram  is  f  of  that 
of  a  square  of  equal  perimeter.     Find  the  perpendicular  distance 
between  each  pair  of  parallel  sides. 

5.  The  lengths  of  the  parallel  sides  of  a  trapezoid  are  12  ft. 
and  17  ft.,  and  the  perpendicular  distance  between  these  sides  is 
8  ft.     Find  its  area. 

6.  If  the  parallel  sides  of  a  garden  are  84  and  92  ft.  respec- 
tively, and  their  perpendicular  distance  82^-  ft.,  what  did  it  cost 
at  $  1200  an  A.  ? 

7.  The  area  of  a  trapezoidal  field  is  3J  A.,  and  the  sum  of 
the  lengths  of  the  parallel  sides  is  440  yd.     Find  the  perpen- 
dicular distance  between  these  sides.     The  lengths  of  the  parallel 
sides  being  in  the  ratio  of  5  to  6,  find  these  lengths. 

8.  The  area  of  the  trapezoid  is  9750  sq.  yd.,  and  the  perpen- 
dicular distance  between  the  parallel  sides  is  234   ft.     If   the 
length  of  one  of  the  parallel  sides  be  410  ft.,  what  will  be  the 
length  of  the  other  parallel  side  ? 


304  ARITHMETIC 

289.  Let  ADC  be  a  triangle,  and  let  the  rectangle  BODE  be  drawn. 
Then  it  is  evident  that  the  triangles  a  and  a'  and  c  and  c'  are  equal.      Hence 
the  triangle  ABC  is  one-half  of  the  rectangle  BCDE.     Hence,  to  find  the 
area  of  a  triangle,  multiply  one-half  the  measure  of  the  base  by  that  of  the 
altitude. 

290.  To  find  the  area  of  a  triangle  when  the  lengths  of  the 
sides  are  given  : 

Find  one-half  of  the  sum  of  the  measures  of  the  sides;  subtract 
from  this  the  measure  of  each  side  separately.  The  square 
root  of  the  product  of  these  four  results  will  give  the  measure  of 
the  area  of  the  triangle. 


Exercise  182 

1.  Find  the  area  of  a  triangle  whose  base  is  45  ft.,  and  alti- 
tude 17  ft. 

2.  A  triangular  piece  of  ground  containing  4|  A.  has  a  base 
of  135  yd.     Find  its  altitude. 

Find  the  areas  of  the  triangles  the  lengths  of  whose  sides  are 
respectively : 

3.  13  yd.,  10  yd.,  and  13  yd. 

4.  13  yd.,  24  yd.,  and  13  yd. 

5.  13  ft.,  4  ft.,  and  15  ft. 

6.  13  ft.,  14  ft.,  and  15  ft. 

7.  13  in.,  11  in.,  and  20  in. 

8.  13  in.,  21  in.,  and  20  in. 

9.  1.23  ch.,  5.95  ch.,  and  6.76  ch. 

10.  73.2  ch.,  45.5  ch.,  and  87.6  ch. 

11.  Find  the  number  of  A.,  etc.,  that  there  are  in  a  triangu- 
lar field  of  which  the  sides  are  7  ch.  60  1.,  9  ch.  50  1.,  and  5  ch. 
701. 


MENSURATION 


305 


12.  The  three  sides  of  a  triangle  are  33  ft.,  56  ft.,  and  65  ft. 
Find  the  measure  of  the  triangle  cut  off  by  joining  the  points  of 
bisections  of  the  two  greater  sides. 

13.  Multiply  the  length  of  each  side  of  the  triangle  in  question 
3  by  3.     Find  the  area  of  the  resulting  triangle.     What  is  the 
ratio  of  its  area  to  that  of  the  A  in  question  3  ? 

14.  Similarly,  multiply  each  side  of  the  A  in  question  4  by  5, 
and  find  its  area.     Find  the  area  of  the  resulting  triangle.     What 
is  the  ratio  of  the  areas  ? 

15.  If  the  sides  of  the  triangle  in  question  5  be  multiplied  by 
the  number  6,  what  will  be  the  ratio  of  its  area  to  that  of  the  tri- 
angle in  question  5  ?     Hence  find  its  area. 

16.  If  each  side  of  the  triangle  in  question  7  be  made  10  times 
as  long,  how  much  greater  will  be  the  area  of  the  triangle  ? 

17.  State  in  what  ratio  the  area  of  a  triangle  is  increased  by 
increasing  the  length  of  each  side. 

291.    To  find  the  area  of  a  circle  : 


306 


ARITHMETIC 


Draw  a  circle  on  cardboard  and  cut  it  out  —  the  larger  the  better.  Divide 
each  half  of  the  circle  as  the  semicircle  in  the  figure  is  divided,  the  arcs 
J,  .B,  (7,  D,  etc.,  being  as  nearly  equal  as  possible.  Cut  the  circle  into  two 
equal  parts  along  the  line  AOM. 

Cut  along  0#,  0(7,  etc.,  cutting  nearly  to  the  points  B,  C,  D,  but  not 
separating  the  parts  entirely  at  these  points.  Spread  the  resulting  figure  out 
as  in  the  darker  part  of  the  figure  below. 


0000 


Then  cut  up  the  other  semicircle  in  the  same  way  ;  spread  open  the  parts 
and  fit  the  two  semicircles  together  as  in  the  figure.  The  resulting  figures  will 
be  nearly  a  rectangle.  The  smaller  the  arcs  AB,  BC,  etc.,  the  more  nearly 
the  area  will  be  to  a  rectangle  whose  base  is  equal  to  one-half  the  circumfer- 
ence and  whose  altitude  is  equal  to  the  radius  of  the  circle. 

Hence  the  measure  of  the  area  of  a  circle  is  one-half  the 
product  of  the  measures  of  the  circumference  and  the  radius.  It 
may  also  be  expressed  thus : 

The  measure  of  the  area  =  ^  cr. 
Again,  since  c  =  3.1416  x  2  r, 
the  measure  of  the  area  =  3.1416  r2. 

Both  formulas  are  useful. 

The  last  rule  may  be  read :  The  measure  of  the  area  of 
a  circle  is  found  by  multiplying  the  square  of  the  measure 
of  the  radius  by  3.1416. 


MENSURATION  307 

292.    (1)  Find  the  area  of  a  circle  whose  diameter  is  7  J  in. 

The  measure  of  the  area  =  3.1416  r2. 
The  radius  =  7J  in.  H-  2  =  3.75  in. 
The  measure  of  the  area  =  3.1416  x  3.752  =  44.18. 
.-.  the  area  =  44.18  sq.  in. 

(2)  If  the  arc  of  a  circle  is  2  ft.  and  the  radius  6  ft., 
find  how  many  degrees  there  are  in  the  arc. 

The  length  of  the  circumference  =  3.1416  x  12  ft.  =  37.6992. 
37.6992  ft.  of  circumference  contain    360°. 

360° 
1  ft.  of  circumference  contains . 

37.6992 

720° 
2  ft.  of  circumference  contain or  19°  5'  54". 

37.6992 
.-.  the  arc  contains  19°  5'  54". 

(3)  Find  the  length  of  a  radius  of  a  circle  whose  area 
is  4  A. 

4  A.  =  19,360  sq.  yd. 
The  measure  of  the  area  =  3.1416  r2. 
.-.  r2  =  19,360  -  3.1416  =  6162.46. 


Exercise  183 

1.  Find  the  area  of  a  circle  7  ft.  in  diameter. 

2.  Find  the  area  of  a  quadrant  whose  radius  is  4  rd. 

3.  Find  the  length  of  the  radius  of  a  circle  whose  area  is 
1  A. 

4.  Find  the  length  of  the  diameter  of  a  circle  whose  area 
is  1  sq.  mi. 

5.  Find  the  length  of  the  radius  of  a  circle  whose  area  is 
equal  to  the  sum  of  the  areas  of  four  circles  of  10  in.,  15  in.,  18  in., 
and  24  in.  radius  respectively. 

6.  Find  the  total  pressure  on  a  plate  25  in.  in  diameter,  the 
pressure  per  sq.  in.  being  65  Ib. 


308  ARITHMETIC 

7.  A  circular  hole  is  cut  in  a  circular  metal  plate  of  7  in. 
radius,  so  that  the  weight  of  the  plate  is  reduced  by  40%.     Find 
the  length  of  the  radius  of  the  hole. 

8.  The  area  of  a  semicircle  is  13.1  sq.  in.     Find  the  length 
of  its  perimeter. 

9.  The  lengths  of  the  sides  of  a  triangle  are  13  ft.,  14  ft.,  and 
15  ft.  respectively.     Find  the  difference  between  the  area  of  the 
triangle  and  that  of  a  circle  of  equal  perimeter. 

10.  The  perimeters  of  a  circle,  a  square,  and  an  equilateral  tri- 
angle are  each  17  ft.  in  length.     Find  by  how  much  the  area  of 
the  circle  exceeds  the  area  of  the  other  figures. 

11.  Find  the  length  of    the  diameter  of  a  circle  whose  area 
is  equal  to  that  of  a  square  whose  sides  are  each  12  ft.  long. 

12.  Out  of  a  circle  of  radius  3  ft.  is  taken  a  circle  of  radius 
2  ft.     Find  the  area  of  the  remainder. 

13.  Find  the  length  of  the  arc  which  subtends  an  angle  of  60° 
at  the  centre  of  a  circle  of  10  in.  radius. 

14.  Find  the  length  of  the  arc  which  subtends  an  angle  of  36° 
at  the  centre  of  a  circle  of  25  in.  radius. 

15.  How  many  degrees  are  there  in  the  angle  which  an  arc 
whose  length  is  1  ft.  subtends  at  the  centre  of  a  circle  of  2  ft. 
radius  ? 

16.  The  length  of  the  radius  of  a  circle  is  8  in.      Find  the 
length  of  the  arc  of  which  the  angle  is  (a)  90°,  (b)  270°. 

17.  There  is  a  circular  fish-pond  of  90  ft.  radius,  surrounded 
by  a  walk  25  ft.  in  breadth.     Find  the  area  of  the  walk. 

18.  Within  a  circular   garden  70   ch.   in    circumference  is  a 
circular  pond  70  rd.  in  circumference.      Find  the  width  of  the 
ring  of  land  which  surrounds  the  pond. 

19.  The  radii  of  four  circles  are  respectively  2  ft.,  3  ft.,  4  ft., 
5  ft.     Show  that  their  areas  are  as  the  numbers  4,  9,  16,  and  25. 


MENSURATION  309 

293.  If  we   wrap  a   rectangular  sheet   of   paper   about  a 
cylinder,  we  find  that  the  area  of  the  curved  surface  of  the 
cylinder  is  a  rectangle  whose  base   is   the   circumference   of 
the  cylinder  and  altitude  the  height  of  the  cylinder.     Hence 
we  can  find  its  area. 

294.  To  find  the  measure  of  the  volume  of  a  cylinder,  take 
the  product  of  the  measures  of  the  area  of  the  base  and    the 
altitude. 

295.  The  curved  surface  of  a  cone  can  be  unwrapped  into 
a  portion  of  a  circle. 


Hence   the    measure  of  the  curved  surface   is   one-half  the 
product  of  the  measure  of  its  base  by  its  slant  height. 

296.  Make  a  cylinder  out  of  paper  and  also  a  right  circular 
cone  having  the  same  altitude  and  base.  Fill  the  cone  with 
some  dry  material  and  empty  it  into  the  cylinder.  Do  this 
three  times  and  the  cylinder  will  be  just  filled.  Hence  the 
volume  of  a  right  circular  cone  is  one-third  that  of  a  cylinder 
of  equal  base  and  altitude.  Hence,  to  find  the  volume  of  a 
right  circular  cone,  multiply  one-third  the  measure  of  the  area 
of  the  base  by  the  measure  of  the  altitude. 


310 


ARITHMETIC 


297.  (1)  The  length  of  the  radius  of  a  right  circular 
cylinder  is  5  in.  and  its  altitude  is  8  in.  Find  its  volume 
and  the  area  of  its  curved  surface. 

The  measure  of  the  area  of  the  base  =  3.1416  x  25  =  78.54. 
The  measure  of  the  volume  of  the  cylinder  =  8  x  78.54  =  628.32. 

.-.  the  volume  =  628.32  cu.  in. 

Again, 

The  measure  of  the  circumference  of  the  base  =  3.1416  x  10  =  31.416. 
The  measure  of  the  area  of  the  curved  surface  =  8  x  31.416  =  251.328. 

.-.  the  area  =  251.328  sq.  in. 

(2)  Find  the  area  of  the  curved  surface  and  also  the 
volume  of  a  cone  whose  altitude  is  8  in.  and  whose  base  is 
12  in.  in  diameter. 


10 


Since  the  altitude  AC  is  perpendicular  to  the  diameter  BCD,  the  triangle 
ACS  is  a  right  triangle. 
Hence  AB  is  10  in. 

Also  the  circumference  of  the  base  =  3.1416  x  12  in.  =  37.6992  in. 
The  measure  of  the  area  =  |  x  \°  x  37.6992  =  188.496. 

/.  the  area  =  188.496  sq.  in. 

Again, 

The  altitude  of  the  cone  =  8  in. 

The  area  of  the  base  =  3.1416  x  62  or  113.0976  sq.  in. 
The  measure  of  the  volume  =  i  x  8  x  113.0976  -  301.5936. 

.-.  the  volume  =  301.59  cu.  in. 


MENSURATION  311 

Exercise  184 

1.  Make  a  cone  and  cylinder  each  of  whose  bases  is  9  in.  in 
circumference  and  altitudes  6  in. 

Fill  the  cone  and  empty  it  three  times,  in  succession,  into  the 
cylinder.     What  is  the  result  ? 

2.  Find  the  volume  of  a  cylinder  the  radius  of  whose  base  is 
10  in.,  the  altitude  being  18  in. 

3.  Find  the  volume  of  a  cone  the  radius  of  whose  base  is  10 
in.,  the  altitude  being  18  in. 

4.  How  often  can  the  cone  in  question  3  be  filled  and  emptied 
into  the  cylinder  in  question  2  ? 

5.  The  length  of  the  radius  of  the  base  of  a  right  circular 
cylinder  is  9  in.  and  its  altitude  is  16  in.     Find  the  volume. 

6.  Find  the  area  of  the  curved  surface  of  the  cylinder  in 
question  5.     Find  the  area  of  its  entire  surface. 

7.  Find  the  volume  of  a  cone  whose  altitude  is  15  in.,  and 
whose  base  is  a  circle  10  in.  in  diameter. 

8.  Find  the  volume  of  a  cone  whose  altitude  is  12  in.,  and  the 
diameter  of  whose  base  is  5  in. 

9.  Find  the  area  of  the  curved  surface  of  a  cone  whose  alti- 
tude is  20  in.,  and  the  radius  of  whose  base  is  15  in.     Find  also 
its  total  area. 

10.  What   must   be   the   height    of   a   cylindrical    column    of 
marble,  the  radius  of  whose  base  is  9  in.,  in  order  that  it  may 
contain  5^  cu.  ft.  ? 

11.  If  the  diameter  of  a  cylindrical  well  be  5  ft.,  and  its  depth 
27  ft.,  how  many  cu.  yd.   of  earth  were  removed  in  order  to 
form  it  ? 

298.    It  can  be  shown  that  the  area  of  the  curved  surface  of 
a  hemisphere  is  equal  to  twice  the  area  of  its  flat  surface ; 


312  ARITHMETIC 

hence  the  area  of  the  surface  of  the  sphere  is  equal  to  4 
times  the  area  of  this  flat  surface. 

Thus  the  measure  of  the  area  of  the  surface  of  a  sphere  is  4  times  the 
product  of  3.1416  and  the  square  of  the  measure  of  the  length  of  the  radius. 

A  =  4:  x  3. 1416  r2. 

299.  If  we  imagine  a  sphere  to  be  divided  into  a  large 
number  of  small  cones,  as  in  §  291  we  divided  the  circle 
into  triangles,  the  centre  of  the  sphere  being  the  vertex  of 
each  cone,  and  a  small  portion  of  the  circumference  being 
its  base,  we  can  think  of  the  volume  of  the  sphere  as  being 
equal  to  the  sum  of  the  volumes  of  the  cones.     The  altitude 
of  each  cone  is  equal  to  the  radius  of  the  sphere,  and  the 
total  area  of  their  bases  is  equal  to  the  area  of  its  surface. 
Hence  the  volume  of  the  sphere  is  given  by  the  formula : 

F=lr(4  x  3.1416  x  r2) 
=  |x  3. 1416  r3. 

Hence  the  measure  of  the  volume  of  the  sphere  is  |-  0/3.1416 
times  the  cube  of  the  measure  of  the  radius. 

300.  (1)  Find  the  surface  of  a  sphere  whose  radius  is  6  in. 

The  measure  of  the  area  =  4  x  3.1416  x  62  =  452.3904. 
.-.  the  area  =  452.39  sq.  in. 


(2)  Find  the  volume  of  a  sphere  whose  diameter  is  8  in. 

the  volume  =  f  x  3.1416  x  < 
the  volume  =  268.08  cu.  in. 


The  measure  of  the  volume  =  f  x  3.1416  x  43  =  268.0832. 


Exercise  185 

1.  Find  the  surface  of  a  sphere  whose  radius  is  3  in. 

2.  Find  the  surface  of  a  sphere  12  in.  in  diameter. 

3.  Find   the   volumes   of  the   spheres    given   in   questions    1 
and  2. 


MENSURATION  313 

4.  Find  the  surface  of  a  sphere  5  ft.  in  diameter. 

5.  Find  the  volume  of  a  sphere  whose  diameter  is  16  ft. 

6.  Place   a   croquet   or   base  ball   between  two  chalk   boxes. 
Place  a  foot  measure  in  line  with  one  edge  of  each  box.     What  is 
the  diameter  of  the   ball  ?     What  is  the  area  of   its  surface  ? 
What  is  its  volume  ? 

7.  With  a  pair  of  compasses  draw  a  circle  with  the  diameter 
found  in  question  6.    Cut  out  this  circle  and  pass  the  ball  through 
the  hole. 

8.  If  the  pressure  of  the  air  is  equal  to  15  Ib.  a  sq.  in.,  what 
is  the  pressure  on  the  surface  of  a  sphere  6  in.  in  diameter  ? 


CHAPTER   XVIII 

THE  METEIO  SYSTEM  OF  WEIGHTS  AND  MEASUKES 

301.  The  French  or  Metric  System  of  Weights  and  Meas- 
ures is  based  upon  the  decimal  system.     It  is  used  in  scien- 
tific treatises  and  has  been  adopted  by  most  of  the  nations  of 
Europe  and  South  America.     It  is  also  in  partial  use  in  the 
United  States  and  Canada. 

• 

302.  The  fundamental  unit  of   the  metric  system  is  the 
Meter,  which  is  39.37  in.  long.     The  original  standard  meter 
is  a  platinum  rod,  called  the  French  Standard  Meter,  which 
is  deposited  in  the  Archives  at  Paris. 

FOUR   INCHES   IN    SIXTEENTHS  OF  AN  INCH 


1 

1 

1 

1 

1 

I 

I 

I 

I 

I 

I 

I 

\ 

I 

I 

I 

I 

! 

I 

1 

( 

I 

j 

3 

4 

1 

2 

f 

\ 

t 

t 

c 

6 

r 

8 

9 

1 

0 

i 

1 

1 

II 

II 

| 

| 

|| 

|| 

| 

I 

|| 

| 

| 

| 

! 

\ 

| 

| 

| 

| 

i 

ONE    DECIMETER    IN    MILLIMETERS 

The  length  of  the  measure  in  the  diagram  is  -^  of  a 
meter,  and  is  called  a  decimeter.  It  is  divided  into  10  equal 
parts,  each  of  which  is  called  a  centimeter.  Each  of  these  is 
divided  into  10  equal  parts,  each  of  which  is  called  a  milli- 
meter. 

303.  In  order  that  pupils  may  study  this  system  of  weights  and  meas- 
ures to  the  best  advantage,  the  school  should  be  provided  with,  a  system  of 

314 


METRIC   SYSTEM  315 

metric  weights  and  measures,  and  each  pupil  with  a  foot  rule  on  which  the 
decimeter,  centimeter,  and  millimeter  are  marked. 

An  intelligent  study  of  the  system  can,  however,  be  made  if  the  teacher 
has  a  metric  stick  and  a  liter  for  reference. 

304.  The  names  of  the  higher  or  lower  units  in  the  metric 
system  are  formed  by  attaching  certain  prefixes  to  the  names 
of  the  standard  units,  thus : 

Deca  signifies  10  times  the  unit. 
Hecto  signifies  100  times  the  unit. 
Kilo  signifies  1000  times  the  unit. 

Deci  signifies  the  10th  part  or  .1  of  the  unit. 
Genii  signifies  the  100th  part  or  .01  of  the  unit. 
Milli  signifies  the  1000th  part  or  .001  of  the  unit. 

UNITS  OF  LENGTH 

1  millimeter  (mm.)    =  .001  meter 
1  centimeter  (cm.)     =  .01  meter 
1  decimeter  (dm.)      =  .1  meter 
1  meter  (m.)  =  Standard  unit 

1  decameter  (Dm.)    =  10  meters 
1  hectometer  (Hm.)  =  100  meters 
1  kilometer  (Km.)     =  1000  meters 
1  rnyriameter  (Mm.)  —  10,000  meters 

The  units  in  common  use  are  the  millimeter,  centimeter,  meter,  and  kilo- 
meter. 

305.  Write  out  the  table  of  the  units  of  length,  thus : 

(a)  10  millimeters  =  1    centimeter,  etc. 

(b)  1  millimeter    =  -^  centimeter,  etc. 

Exercise  186 

1.  Measure  with  a  yardstick  in  the  school-yard  a  distance 
of  11  yd.  Measure  along  the  same  distance  10  times  with  the 
meter.  What  is  the  difference  in  inches  ? 


316  ARITHMETIC 

2.  Keduce  11  yd.  and  also  10  m.  to  in.,  and  verify  the  result 
obtained  in  the  first  question. 

3.  Measure  the  length  of  the  schoolroom  in  m.  and  decimals 
of  a  m. 

4.  Find  the  hypotenuse  of  a  right  triangle  whose  sides  are 
(1)  6  m.  and  8  in.,  (2)  24  cm.  and  45  cm. 

5.  Librarians  frequently  use  the   centimeter   as  the  unit  in 
registering  the  heights  of  books.     Express  in  cm.  the  height  of 
your  (a)  Arithmetic,  (b)  History,  (c)  Geography,  (d)  and  also  of 
several  other  books. 

6.  Measure  and  express  in  terms  of   the  cm.  the  distances 
between  the  lines  on  ruled  paper. 

7.  Measure  and  express  in  terms  of  the  cm.  the  height  of  the 
schoolroom  thermometer. 

8.  How  many  cm.  are  there  in  a  full  line  of  this  book  ? 

9.  Press  tightly  together  the  leaves  of  this  book.     Make  a 
layer  just  1  cm.  thick.     Count  the  leaves  and  find  the  thickness 
of  1  leaf  as  a  decimal  of  a  mm. 

10.  Cut  a  slit  2  mm.  wide,  by  3  cm.  long,  in  a  sheet  of  paper. 

11.  How  many  mm.  are  there  in  the  width  of  your  pencil  ? 

12.  Find  the  number  of  in.  in  1  Km.,  reduce  the  result  to  a 
decimal  of  a  mi.,  and  show  that  1  Km.  is  nearly  equal  to  -f 
of  a  mi. 

13.  If  a  train  travels  at  the  rate  of  20  m.  a  sec.,  what  is  the 
rate  in  Km.  per  hr.  ? 

14.  Show  that  5  in.  are  very  nearly  equal  to  127  mm. 

15.  For  what  do  we  use  the  in.,  the  yd.,  and  the  mi.  ?     For  what 
are  the  mm.,  the  cm.,  the  m.,  and  the  Km.  respectively  used  ? 


METRIC   SYSTEM  317 

UNITS  OF  AREA 

306.  The  principal  units  for  measuring  land  are  the  square 
meter,  called  the  Centare  (ca.),  the  square  decameter  called  the 
Are  (a.),  and  the  square  hektometer  called  the  Hectare  (ha.). 


1  sq. 
cm. 


1  sq.  dm. 


Each  side  of  this  square  measures 

1  dm.,  or 

3if  in.,  very  nearly. 

A  liter  is  a  cube  each  face  of  which  has  the  dimensions  of  this 
square. 

A  gram  is  the  weight  of  a  cubic  centimeter  (see  small  square 
above)  of  distilled  water,  weighed  in  vacuo  at  temperature  of 
maximum  density,  39.1  F.  A  liter  or  cubic  decimeter  of  such 
water  weighs  1  kg.  or  2£  Ib.  nearly. 


307.  How  many  units  of  length  are  there  in  the  side  of 
a  square  meter,  the  decimeter  being  the  unit? 

How  many  units  of  area  are  there  in  a  square  meter,  the 
square  decimeter  being  the  unit? 


318  ARITHMETIC 

How  many  units  of  area  are  there  in  a  square  decimeter, 
the  square  centimeter  being  the  unit  of  area  ? 

What  part  of  a  square  meter  is  a  square  decimeter? 

Write  a  square  decimeter  as  a  decimal  of  a  square  meter. 

Write  a  square  centimeter  as  a  decimal  of  a  square  deci- 
meter. As  a  decimal  of  a  square  meter.  What  is  the  ratio 
of  each  unit  of  area  in  the  metric  system  to  the  next  smaller, 
and  also  to  the  next  higher? 

UNITS  OF  AREA 

1  square  millimeter  (q.  mm.)  =  .000001  of  a  square  meter 
1  square  centimeter  (q.  cm.)  =  .0001  of  a  square  meter 
1  square  decimeter  =  .01  of  a  square  meter 

1  square  meter  (q.  m.)  =  standard  unit 

1  square  decameter  =  100  square  meters 

1  square  hektometer  —  10,000  square  meters 

1  square  kilometer  (q.  Km.)  =  1,000,000  square  meters 
1  centare  (ca.)  =  .01  are  (a.) 

1  hectare  (ha.)  =  100  ares 

Note  that  the  square  meter  is  1^  times  as  large  as  the  square  yard. 

Exercise  187 

1.  Cut  out  of  paper  a  q.  dm.  and  also  a  q.  cm.     What  is  the 
ratio  of  the  two  areas  ? 

2.  Draw  a  sq.  ft.  and  measure  it  with  the  q.  dm.  of  paper  as 
the  unit  of  area. 

3.  A  q.  dm.    contains  .10764   sq.  ft.      Find   the   number   of 
q.  dm.  contained  in  a  sq.  ft.  correct  to  two  decimal  places,  and 
compare  the  result  with  that  obtained  in  question  2. 

4.  Make  a  drawing  of  a  q.  m.,  and  draw  a  sq.  yd.  within  it. 

5.  Mark  out  an  are  on  the  school-ground. 

6.  State  the  table  of  area,  expressing  each  unit  of  area  as 
equal  to  100  times  the  next  lower. 

7.  Find  the  area  of  a  page  of  this  book  in  q.  cm. 

8.  Find  the  area  of  the  room  in  ca. 


METRIC   SYSTEM  319 

UNITS  OF  VOLUME 

308.  The  principal  units  of  volume  are  the  cubic  meter, 
also  called  the  Stere,  and  the  cubic  decimeter,  called  the  Liter. 

309.  How  many  units  of  length  are  there  in  a  side  of 
a  meter,  the  decimeter  being  the  unit  ? 

How  many  units  of  volume  are  there  in  the  cubic  meter 
or  stere,  the  cubic  decimeter  or  liter  being  the  unit  ? 

What  is  the  ratio  of  each  unit  of  volume  in  the  metric 
system  to  the  next  smaller  unit?  To  the  next  larger? 

UNITS  OF  VOLUME 

1  cubic  millimeter  (c.  mm.)=  .000000001  of  a  cubic  meter 
I  cubic  centimeter  (c.  cm.)   =  .000001  of  a  cubic  meter 
1  cubic  decimeter  =  .001  of  a  cubic  meter 

1  cubic  meter  (c.  m.)  =  standard  unit 

Exercise  188 

1.  Make  a  liter  out  of  paper. 

2.  Fill  a  qt.,  liquid   measure,  with  sand  and  empty  it  into 
your  liter.     Which  of  the  two  measures  is  larger  ? 

3.  Fill  a  liter  with  sand  and  empty  it  into  a  qt.,  dry  meas- 
ure.    Which  of  the  two  is  larger  ? 

4.  Fill  a  gal.  measure,  using  the  liter  as  a  dipper,  and  note 
how  many  liters  are  equivalent  to  the  gal. 

5.  If  1  liter  is  equal  to  .264  gal.,  find  correct  to  two  decimal 
places  the  number  of  liters  in  a  gallon.      Compare  this  result 
with  that  obtained  in  question  4. 

6.  State  some  purposes  for  which  the  liter  is  used. 

7.  Make  a  c.  cm.  out  of  paper. 

8.  Make  the  necessary  measurements  and  compute  the  volume 
of  the  room  in  c.  m. 


320  ARITHMETIC 

9.    Make  the  necessary  measurements  and  compute  the  volume 
of  a  box  in  liters. 

10.  Express  as  a  decimal  part  of  a  c.  m.  the  volume  of  a  beam 
3  m.  long,  10  cm.  wide,  and  5  cm.  thick. 

11.  A  cylindrical  vessel  having  a  base  of  a  q.  m.  is  filled  with 
water  to   the  depth  of   2  m.     How  many  liters  of  water  does 
it  contain? 

12.  How  many  liters  of  water  may  be  held  by  a  vessel  meas- 
uring 25  x  35  x  75  cm.  ? 

13.  What  will  it  cost  to  build  a  wall  1  Hm.  long,  1  dm.  thick, 
and  1  m.  high,  at  $  5  a  c.  m.  ? 

UNITS  OF  WEIGHT 

310.  The  principal  units  of  weight  are  the  Gram  and  the 
Kilogram. 

The  Gram  is  the  weight  of  a  cubic  centimeter  of  distilled 
water  at  40°,  at  which  temperature  water  is  at  its  maximum 
density. 

A  nickel  weighs  5  g. 

A  liter  of  distilled  water  at  40°  weighs  1  kg.  The  kilogram  is  nearly 
equal  to  "2\  Ib.  Avoir. 

A  cubic  meter  of  water  at  40°  weighs  a  metric  ton  (1000  kg.). 

Exercise  189 

1.  One   gram  is   equal  to   15.432  gr.      Show  that  1   kg.   is 
approximately  equal  to  2-J-  Ib.  Avoir. 

2.  A  cubic  meter  of  distilled  water  at  40°  weighs  how  many 
kg.  ?     If  1  kg.  is  equal  to  21  Ib.  Avoir.,  how  many  pounds  does 
1  c.  m.  of  water  weigh  ? 

3.  Show  that  a  metric  ton  weighs  about  10%  more  than  our 
short  ton. 


METRIC   SYSTEM  321 

4.  If  sulphuric  acid  is   1.8  times  as  heavy  as  water,  what 
weight  of  the  acid  will  a  two-liter  bottle  contain  ? 

5.  What  part  of  a  liter  is  750  g.  of  water  ? 

6.  What  is  the  weight  of  1  deciliter  of  water? 

7.  If  alcohol  is  80%  as  heavy  as  water,  what  will  375  c.  cm. 
of  alcohol  weigh  ? 

8.  If  20  c.  cm.  of  lead  weighs  227  g.,  what  is  the  ratio  of  the 
weight  of  lead  to  that  of  an  equal  volume  of  water  ? 

9.  If  a  quantity  of  iron  weighs  7.8  times  as  much  as  an  equal 
quantity  of  water,  what  is  the  weight  of  an  iron  bar  75  x  4  x  3  cm.  ? 

10.  A  body  weighing  512  g.  in  air  weighs  428  g.  in  water. 
What  per  cent  of  its  weight  is  lost  ? 

11.  A  liter  flask  was  two-fifths  filled  with  water;  the  remain- 
ing space  being  filled  with  sand,  the  weight  was  found  to  be 
2050  g.     Required  the  weight  of  a  liter  of  sand. 

12.  If  the  pressure  of  the  air  on  the  surface  of  a  body  is  1  kg. 
to  the  q.  cm.,  what  is  the  pressure  of  the  air  on  the  surface  of  a 
sphere  whose  radius  is  10  cm.  ? 

13.  A   cubical  block  of   ice  measures   3  dm.   along   its   edge. 
What  will  be  its  weight  if  ice  weighs  94%  as  much  as  an  equal 
volume  of  water  ? 

14.  What  is  the  weight  of  air  in  a  room  5  m.  long,  3  m.  wide, 
4  m.  high,  if  1  c.  dm.  of  air  weighs  .0018  kg.  ? 

Y 


CHAPTER   XIX 

Miscellaneous  Exercise  190 

1.  Write  the  following  in  figures  : 

(a)  Fifty  thousand  nine  hundred  and  nine. 

(5)  Nine  hundred  thousand  and  ninety. 

(c)    Six  hundred  and  fifty  thousand  seven  hundred. 

(cf)  Eight  hundred  and  seven  thousand  and  eight. 

(e)    Seven  hundred  and  seventy  thousand  and  sixty-seven. 

(/)  Nine  million  ninety  thousand  and  ninety-nine. 

(g)   Eighty  million  nine  hundred  thousand  and  thirty. 

(7i)  Nine  hundred  and  seventy ,  million  eight  hundred  and 
eighty-seven  thousand. 

(i)    Six  hundred  and  seventeen  million  and  ninety-three. 

(j)  Nine  hundred  and  nineteen  thousand  four  hundred  and 
eleven. 

2.  Write  in  figures  : 

Twenty -five  thousand  four  hundred  and  ninety ;  ninety-nine 
thousand  nine  hundred  and  seventeen ;  nine  hundred  and  seven 
thousand  six  hundred  and  six ;  one  million  ;  MDCCCXC  V.  And 
in  words  :  9009 ;  16,060  ;  7018  ;  207,509 ;  75,115. 

3.  (a)  Define  and  give  examples  of  quantity,  unit,  and  number. 

(6)  Explain  the  basis  of  our  system  of  numeration. 

4.  Write  in  figures  (placed  for  addition) :  Nine  hundred  and 
nineteen  ;  three  hundred  and  eleven ;  seven  hundred  and  seventy  ; 
eight  hundred  and  ninety-seven  ;  six  hundred  and  eight ;  XCVII. ; 
LXVII. ;  CXIX. ;  CDL. ;  and  DCXL. 

322 


MISCELLANEOUS   EXERCISE 


323 


5.  Add:  4567890123 

5678901234 
6789012345 
7890123456 
8901234567 
0912345678 
6598695326 
8396876549 
7788995566 
3453453456 

6.  Write  down  neatly  the  following  statement  of  six  weeks' 
cash  receipts ;  add  the  amounts  vertically  and  horizontally,  and 
prove  the  correctness  of  the  work  by  adding  your  results : 


MON. 

TUBS. 

WED. 

THUR. 

FRI. 

SAT. 

TOTAL 

1st 

$28.79 

$34.71 

$  35.33 

$  30.10 

$27.97 

$47.81 

2d 

23.87 

30.03 

29.38 

33.84 

26.77 

48.77 

3d 

16.99 

27.09 

28.77 

30.16 

24.95 

43.07 

4th 

29.13 

33.72 

30.81 

39.17 

28.47 

50.05 

5th 

18.47 

32.29 

26.73 

34.45 

28.88 

54.39 

6th 

19.02 

27.0G 

29.04 

29.89 

29.51 

61.93 

Total 

7.    Solve,  as  in  question  6 


Mox. 

TUBS. 

WED. 

TllUR. 

FRI. 

SAT. 

TOTAL 

1st 

$  65.95 

$24.89 

$  79.79 

$40.78 

$  37.59 

$89.61 

2d 

58.71 

41.65 

24.67 

94.26 

70.26 

42.51 

3d 

47.58 

99.57 

50.60 

80.71 

91.82 

89.76 

4th 

29.69 

70.80 

87.91 

74.93 

36.63 

21.90 

5th 

81.45 

56.93 

54.82 

96.57 

12.72 

96.67 

6th 

42.63 

68.77 

81.79 

60.86 

31.87 

75.82 

Total 

324  ARITHMETIC 

• 

8.  Mr.  Jones  bought  one  house  for  $865  and   another  for 
$984,  and  sold  them  both  for  $1900.     How  much  did  he  gain? 

9.  John  has  $149,  Will  has  $87  more  than  John,  and  Sam 
has  $  115  more  than  both.     How  many  dollars  has  Sam  ? 

10.  Ned  sold  two  bags  of  potatoes.     With  the  money  he  got 
for  them  he  bought  a  knife  for  25^,  a  saw  for  65^,  a  hatchet  for 
75^,  and  had  45^  left.     How  much  did  he  get  for  his  potatoes  ? 

11.  Two  men  together  receive  $  97.75,  but  one  receives  $  18.25 
more  than  the  other.     How  much  does  each  receive  ? 

12.  A  and  B  start  together  and  walk  in  the  same  direction,  A 
at  the  rate  of  4  mi.  an  hr.,  and  B  at  the  rate  of  3  mi.  an  hr.     At 
the  end  of  7  hr.  A  turns  and  goes  back.      How  many  mi.  will 
B  have  gone  when  he  meets  A  ? 

13.  In  a  factory,  12  men,  16  women,  and  30  boys  are  employed. 
At  the  end  of  a  week  they  receive  $  330.     A  man  is  paid  as 
much  as  2  women,  and  a  woman  as  much  as  3  boys.     What  is  the 
share  of  each  ? 

14.  A  man  bought  a  number  of  cows  for  $1080;  he  sold  half 
of  them  for  $  810,  thereby  gaining  $  15  on  each  one  sold.     What 
did  each  cow  cost  ? 

15.  A  clerk  received  a  salary  of  $650  a  yr.     He  spent  50^ 
a  da.  the  first  yr.,  $  4  a  wk.  the  second  yr.,  and  $  22  a  mo.  the 
third  yr.     How  much  did  he  save  in  three  years  ? 

16.  The  subtrahend  is  9564,  the  remainder  is  1965.     What  is 
the  minuend?     The  multiplier  is  96  and  the  product  is  82,848. 
What  is  the  multiplicand  ? 

17.  The  dividend  is  1800,  the  quotient  is  17,  and  the  remainder 
66.     What  is  the  divisor  ? 

18.  How  many  times  can  506  be  subtracted  from  the  product 
of  6072  and  13,986  ? 

19.  The  quotient  of  a  division  is  834.     What  quotient  would 
have  been  obtained  if  both  dividend  and  divisor  had  been  first 
multiplied  by  13  ?     Why  ? 


MISCELLANEOUS   EXERCISE  325 

20.  Subtract  847^|  from  1003yV,  explaining  fully  each  step. 

21.  Simplify  ^  -  -  f  of  f  +  y7§,  and  find  how  many  times  the 
result  is  contained  in  f  -=-  (•£  of  T3T  —  4). 

O  \  y  1   -r  O  / 

22.  Divide  the  sum  of  |  of  8|  and  21  of  5|  by  the  difference 


between  f  of  3^  and  J  of  i  of  2f  . 


23.  Prove:    (l)|of|  =  1*si    (2)  |  of  |  =  f  of 

24.  Simplify    3J  +|  -        of  3|  -- 


f  . 


25.  A  boy's  age  now  is  i  of  his  father's.     In  6  yr.  it  will  be 
i  his  father's  present  age.     How  old  is  he  ? 

26.  A  house  and  lot  are  together  worth  $  2100;  i  of  the  value 
of  the  house  is  equal  to  -J-  of  the  value  of  the  lot.      Find  the 
value  of  each. 

27.  The  circumference  of  a  wheel  is  -2T2-  of  its  diameter.     Find 
the  diameter  of  a  wagon  wheel  which  makes  360  revolutions  in 
going  a  mile. 

28.  A  man  owned  a  ^-interest  in  a  mill,  and   sold  |-  of  his 
interest  to  one  man,  and  1  of  his  interest  to  another.     What  part 
of  the  mill  did  each  of  the  three  men  then  own  ? 

29.  If  to  a  certain  number  its  J,  -J,  and  i  be  added,  the  sum 
will  be  122  ;  required  the  number. 

30.  Find  the  number  which  is  207  more  than  the  sum  of  -J-  and 
-i-  of  itself. 

31.  A  man  spent  -f$  of  his  money  for  a  house,  f  of  the  remain- 
der for  cattle,  and  the  rest  for  a  farm.     If  the  farm  cost  him 
$  357  less  than  the  house  and  cattle  together,  what  did  he  pay 
for  all? 

32.  A  legacy  of  $9500  is  to  be  divided  among  A,  B,  and  C, 
so  that  A  will  get  T5g-  of  the  whole,  and  B  will  get  f  as  much  as 
0.     Find  the  shares  of  each. 

33.  A   man   spent  ^  of  his   money   for  provisions,   |  of  the 
remainder  for  clothing,  T2^  of  the  remainder  for  charity,  and  had 
f  9.10  left.     How  much  did  he  have  at  first  ? 


326  ARITHMETIC 

34.  Nathan  Curd  sells  a  merchant  752  Ib.  of  cheese  at  11 1 
per  Ib.,  and  receives  the  following  goods  in  exchange : 

'     11  yd.  silk  @  $  2.25 ;  96  Ib.  nails  @  3f  ^ ; 

400  Ib.  sugar  @  4  J^ ;  56  yd.  gray  cotton  @  9f  ^ ; 

12  Ib.  raisins  @  11^  ;  11  yd.  white  cotton  @  10^ ; 

3  pr.  gloves  @  75^. 

Find  the  balance  due  Nathan  Curd. 

35.  A  man  owns  a  horse  and  saddle;  \  of  the  value  of  the 
horse  is  equal  to  4  times  the  value  of  the  saddle ;  the  horse  and 
saddle  together  are  worth  $  170.     Find  the  value  of  each. 

36.  A  man  bought  a  horse  and  carriage  for  $  280,  and  f  of 
the  cost  of  the  carriage  was  equal  to  -§-  of  the  cost  of  the  horse. 
What  was  the  cost  of  each  ? 


37.  Divide  the  product  of  .037  and  .0025  by  the  sum  of  .9,  .02, 
and  .005. 

38.  Divide  6  by  .000725,  correct  to  four  decimal  places. 

39.  Add  together  1.302,  3.2589,  and  40.93.     Multiply  the  sum 
by  .00297  and  divide  the  product  by  90.09. 

40.  Multiply  350.4  by  .0105  and  divide  the  product  by  .0000219. 

41.  What  decimal  must  be  taken  from  the  sum  of  69 J-,  8.2, 
5.445,  .065,  and  20T1-2-,  so  that  it  will  contain  6.05  an  exact  number 
of  times  ? 

42.  A  drover  lost  .065  of  his  flock  by  wolves,  .105  by  disease, 
and  .27  by  theft.     He  then  sold  .75  of  what  remained,  and  has 
280  sheep  left.     Find  the  number  in  his  original  flock. 

43.  Find  the  amount  of  the  following  bill : 

1328  ft.  siding,  at  $  1.621  per  C. ; 
48,480  cu.  ft.  timber,  at  $  59.371  per  M. ; 
7400  fence  rails,  at  $  7.75  per  C.  ; 
8400  fence  pickets,  at  $  15.00  per  M. ; 
5680  Ib.  hay,  at  $  12.50  per  T. 


MISCELLANEOUS   EXERCISE  327 

44.    A  cooper  paid  f  78.32  for  16,488  bbl.  staves.      Kequired 
the  price  per  M. 


45.  A  rectangular  field  is  7  ch.  75  1.  long  and  4  ch.  87-J-  1.  wide. 
How  many  rd.  of  fencing  are  required  to  enclose  it  ? 

46.  How  many  mi.  of  road,  3  rd.  wide  will  contain  8  A.  of 
land? 

47.  Make  a  drawing  that  will   show  the  number  of  sq.   yd. 
in  a  sq.  rd. 

48.  Find  the  value  of  a  piece  of  land  20  ft.  x  40  rd.,  at  $  1000 
per  A. 

49.  A  certain  map  is  drawn  on  a  scale  of  8  mi.  to  an  inch.    On 
this  map  the  township  of  Scott  measures  1T5^  in.  in  length  and 

11  in.  in  width.     How  many  A.  does  it  contain  ? 

50.  Find  the  expense  of  sodding  a  plot  of  ground  which  is 
40  yd.  long  and  100  ft.  wide,  with  sods  each  1  yd.  in  length  and 
1  ft.  in  breadth,  the  sods,  when  laid,  costing  75^  per  hundred. 

51.  A  floor  16  ft.  8  in.  by  14  ft.  2  in.  is  to  be  laid  with  square 
tiles.     Find  the  dimensions  of  the  largest  tiles  that  can  be  used 
without  cutting  or  fitting. 

52.  Find  the  cost  of  papering  a  room  24  ft.  long,  21  ft.  wide, 

12  ft.  high,  at  25^  a  roll,  12  yd.  long  and  21  in.  wide. 

53.  How  much  will    it  cost  to  plaster  the  walls  and  ceiling 
of  a  room  15  ft.  long,  12  ft.  wide,  and  11  ft.  high,  at  32±fl  per 
sq.  yd.  ? 

54.  A  room  18  ft.  by  16  ft.  is  carpeted  with  carpet  j  yd.  wide, 
and  the  smallest  possible  number  of  yd.  of  the  carpet  is  used. 
Find  (a)  the  number  of  breadths,  (b)  the  number  of  yd. 

55.  How  many  thousand  shingles,  18  in.  long  and  4  in.  wide, 
lying  1  to  the  weather,  are  required  to  shingle  the  roof  of  a  build- 


328  ARITHMETIC 

ing  54  ft.  long,  with  rafters  22  ft.  long,  the  first  row  of  shingles 
being  double  ? 

56.  A  schoolroom  is  30  ft.  long,  24  ft.  wide,  and  10  ft.  high 
above  the  wainscoting.     The  trustees  pay  $  20  per  M.  for  a  new 
floor,   $  15  per  M.  for  a  new  board  ceiling,  10^  per  sq.  yd.  for 
painting  the   ceiling,  4^  per  sq.   yd.  for  tinting  the  walls,  and 
$  2  per  da.  for  6  da.  labor.     Find  the  total  cost. 

57.  A  cubical  cistern  is  5  ft.  deep.     How  many  gal.  of  water 
will  it  hold  if  231  cu.  in.  make  a  gal.  ? 

58.  How  many  cubical  blocks,  each  edge  of  which  is  ^  ft.,  are 
equivalent  to  a  block  of  wood   8   ft.  long,  4  ft.  wide,  and  2  ft. 
thick  ? 

59.  If  the  ceiling  of  a  square  room  is  15  ft.  high,  how  many 
sq.  ft.  of  floor  must  it  have  in  order  that  50  pupils   and  the 
teacher  may  each  have  300  cu.  ft.  of  air  ? 

60.  Four-foot  wood  piled  5£  ft.  high  requires  how  many  ft.  in 
length  of  the  pile  for  2J  cd.  ? 

61.  What  is  the  value  of  a  pile  of  wood  360  ft.  long,  12  ft. 
wide,  6  ft.  high,  at  $  3.20  per  cd.  ? 

62.  A  square  plot  of  ground  that  contains  -fo  A.  is  covered 
with  cordwood  (4  ft.  long)  to  an  average  height  of  12  ft.     What 
is  the  wood  worth  at  $  4.12  per  cd.  ? 

63.  Required  the  cost  of   35  pieces  of  scantling,  18  ft.  long, 
4  in.  wide,  and  2  in.  thick,  at  $  14  per  M.,  board  measure. 

64.  How  many  board  feet  are  there  in  12  scantlings  16  ft.  by 
4  in.  by  2  in.  ? 

65.  It  is  required  to  build  a  sidewalk  \  mi.  in  length,  8  ft. 
wide,  and  2   in.  thick,  supported   by  three  continuous  lines   of 
scantling  4  in.  square.     What  will  the  lumber  cost  at  $  17  per  M.  ? 


MISCELLANEOUS   EXERCISE  329 

66.  Find  the  value  of  the  following  lumber  at  $  15  per  M. : 

20  pieces  2  x  4,  18  ft.  long ; 
20  pieces  4  x  4,  12  ft.  long ; 
20  pieces  3  x  10,  16  ft.  long. 

67.  A  farmer  sold  a  lot  of  barley,  weighing  2712   lb.,  when 
barley  was  40j^  per  bu.     In  weighing  the  grain,  the  dealer  made 
a  mistake  and  took  it  as  rye,  and  paid  for  it  at  49^  per  bu.     How 
much  did  the  farmer  gain  or  lose  by  the  mistake  ? 

68.  The  weight  of  a  cu.  ft.  of  water  is  621  lb.,  and  1   gal. 
contains  231  cu.  in.     Find  the  weight  in  oz.  of  1  pt.  of  water. 

69.  A  lake  whose  area  is  45  A.   is   covered  with   ice  3  in. 
thick.     Find  the   weight  of  the  ice  in  T.,  if  1   cu.   ft.   weighs 
920  oz.  Avoir. 

70.  In  what  time  would  a  field,  80  by  60  rd.,  pay  for  under- 
draining  lengthwise,  at  2^  per  ft.,  if  the  field  yield  2  bu.,  at  66^, 
per  A.  more  than  before  draining  ?     The  drains  are  4  rd.  apart, 
and  the  first  drain  runs  down  the  centre  of  the  field. 

71.  Find  the  amount  of  the  following  bill : 

June  1,  1896,  G.  Murray  &  Co.  sold  to  John  Scott,  4886  bu. 
36  lb.  wheat  @  58^  per  bu.,  4532  lb.  peas  @  52^  per  bu.,  38  bu.  3  pk. 
barley  @  54^  per  bu.,  465  lb.  flour  @  $  1.50  per  cwt.,  4685  lb.  bran 
@  $  15  per  T.  Write  out  "a  receipt  in  full  for  payment  of  account, 
June  26. 

72.  Find  the  length  of  the  shortest  line  that  can  be  exactly 
measured  by  a  yard  measure,  a  ten-foot  pole,  or  a  two-rod  chain. 

73.  Kequired  the  cost  of  1  doz.  silver  spoons,  each  weighing 
18  pwt.  18  gr.,  at  $  1.15  per  oz. 

74.  Eeduce  7  gal.  3  qt.  1  pt.  to  the  fraction  of  a  bbl. 

75.  I  sow  11  bu.  2  pk.  4  qt.  of  wheat,  and  raise  therefrom 
215  bu.  2  qt.     How  much  is  the  average  yield  per  bu.  of  seed  ? 


330  ARITHMETIC 

76.  The  running  time  of  the  Empire  State  Express  from  New 
York  to  Buffalo  is  8  hr.  30  min.,  and  the  distance  is  440  mi.     If 
stops  of  5  min.  each  are  made  at  Albany,  Utica,  Syracuse,  and 
Rochester,  what  is  its  average  speed  per  hr.  ? 

77.  Find  (a)  the  exact  number  of  da.  from  Jan.  17,  1896,  to 
April  5,  1896;  (6)  the  difference  in  time  by  subtraction  of  dates. 

78.  A  railroad  train  moves  1  mi.  in  65  sec.     What  is  its  speed 
per  hr.  ? 

79.  A   note   given   Aug.    15,  1896,  for   90   da.,    will   mature 
when  ? 

80.  A  can  walk  3^  mi.  in  50  min.,  and  B  can  walk  21  rni.  in 
36  min.     How  many  yd.  will  A  be  ahead  of  B  when  A  has  gone 
6  mi.,  if  they  start  together  ? 

81.  A  farmer  delivered  at  a  warehouse  four  loads  of  wheat 
weighing  respectively  2113  lb.,  2310  lb.,  2270  lb.,  and  2091  Ib. 
How  much  should  he  have  received  at  72^  per  bu.  ? 

82.  The   difference  in   longitude   between   two   places   being 
9°  34' 25",  what  is  the  difference  in  time  ? 


83.  A  man  has  a  salary  of  $400  a  yr.,  and  has  $500  in  the 
bank.     If  he  spends  $  500  a  yr.,  in  what  time  will  his  money  be 
all  gone  ? 

84.  What  is  the  shortest  stick  that  can  be  cut  into  pieces, 
9  in.,  12  in.,  or  15  in.  in  length,  with  nothing  remaining  ? 

85.  (a)  What  is  meant  by  a  Common  Multiple  of  two  or  more 
fractions  ? 

(b)  Find  the  L.  C.  M.  of  2J,  3|,  3^,  14T\. 

86.  (a)  What  is  meant  by  the  prime  factors  of  a  number  ? 

(b)  Find  the  prime  factors  of  13,230,  22,050,  and  23,625 ;  and 

(c)  By  means  of   the  prime   factors  find  their   G.  .C.  M.  and 
L.  C.  M.' 


MISCELLANEOUS   EXERCISE  331 

87.  Eesolve  16,335  and  18,018  into  their  prime  factors,  and 
from   inspection   of  these   write  the   prime  factors  of  their   (a) 
L.  C.  M.  and  (b)  G.  C.  M. 

88.  A  farmer  bought  a  number  of  horses  and  cows  for  $2000. 
There  were  3  times  as  many  cows  as  horses,  and  a  horse  costs 
twice  as  much  as  a  cow.     If  each  horse  cost  $  80,  how  many 
cows  did  he  buy  ? 

89.  The  difference  in  weight  of  two  chests  of  tea  is  25  Ib.  ;  the 
value  of  both  at  65^  per  Ib.  is  $  113.75.     How  many  Ib.  of  tea  are 
in  each  chest  ? 

90.  What  is  the  smallest  sum  of  money  with  which  you  can 
buy  chickens  at  25^,  or  geese  at  50^,  or  turkeys  at  75^,  or  lambs 
at  $  3,  or  sheep  at  $  5,  or  pigs  at  $  7,  or  cows  at  $  35,  or  horses 
at  $  140,  and  have  exactly  $  15  left  for  expenses  ? 

91.  Ten  cents  will  buy  3  oranges,  4  lemons,  or  5  apples.     How 
many  apples  are  worth  as  much  as  5  doz.  oranges  and  7  doz. 
lemons  ? 

92.  One  workman  charges  $3  for  a  day's  work  of  8  hr.,  and 
another  $  3.50  for  a  day's  work  of  9  hr.     Which  had  I  better 
employ,  and  how  much  shall  I  have  to  pay  him  for  work  that  he 
can  do  in  a  fortnight,  working  6  hr.  a  day  ? 

93.  A  can  do  a  piece  of  work  in  f  of  a  da.,  and  B  in  i  of  a 
da.     In  what  time  can  both  together  do  it?     If  $  1.40  be  paid 
for  the  work,  how  much  should  A  receive  ? 

94.  How  many  oranges  must  a  boy  buy  and  sell  to  make  a 
profit  of  $  9.30,  if  he  buys  at  the  rate  of  5  for  3^,  and  sells  at  the 
rate  of  4  for 


95.  A  and  B  dig  a  ditch  in  50  hr.     With  C's  help  they  could 
have  done  it  in  1S£  hr.     In  what  time  could  C  do  -§-  of  the  work 

o 

alone  ? 

96.  Three  men  can  dig  a  certain  drain  in  8  da.     They  work 
at  it  for  5  da.,  when  one  of  them  falls  ill,  and  the  other  two  finish 


332  ARITHMETIC 

the  work  in  5  da.  more.     How  much  of  the  work  did  the  first 
man  do  before  he  fell  ill  ? 

97.  A  boy  can  run  6  times  around  a  circular  plot  of  ground 
in  52  sec. ;  another  boy  can  run  9  times  around  the  same  plot  in 
80  sec.     If  they  start  from  the  same  place  at  the  same  time,  and 
run  in  the   same   direction,   how  many   rounds   will   each  make 
before  the  faster  boy  overtakes  the  slower  ? 

98.  Express  in  the  form  of  a  vulgar  fraction  the  average  of 

I  A,  .7,  .4f,  and  .4861. 
ij 

99.  In  a  granary  there  are  4  bins,  each  10  ft.  long  and  5  ft. 
wide.     How  high  must  they  be  boarded  in  front  to  be  capable  of 
holding  860  bu.  ? 

100.  The  outfit  of  a  livery  stable  is  worth  $  3000  ;  \  the  value 
of  the  horses  is  equal   to  1  the  value  of  vehicles,  harness,  etc. 
Find  the  value  of  the  horses. 

101.  A  farmer  agreed  to  pay  his  hired  man  10  sheep  and  $  160 
for  1  yr.  labor.     The  man  quit  work  at  the  end  of  7  mo.,  receiving 
the  sheep  and  $  60  as  a  fair  settlement.     Find  the  value  of  each 
sheep. 

102.  Divide  $1200  among  A,  B,  and  C,  so  that  A  may  have 
$  70  more  than  B,  and  twice  as  much  as  C. 

103.  A  train  going  25  mi.  an  hr.  starts  at  1  o'clock  P.M.  on 
a  trip  of  280  mi. ;  another  going  37  mi.  an  hr.  starts  for  the  same 
place  at  12  min.  past  4  o'clock  P.M.     When  and  where  will  the 
former  be  overtaken? 

104.  In  the  number,  28,672,  the  value  expressed  by  the  first 
two  digits  from  the  left  is  how  many  times  the  value  expressed 
by  the  fourth  digit  from  the  left  ? 


105.  A  town  whose  population  was  10,000  increased  10% 
every  year  for  3  yr.  What  was  its  population  at  the  end  of 
that  period  ? 


MISCELLANEOUS   EXERCISE  333 

106.  A  house  and  lot  was  sold  for  $7030,  at  a  loss  of  16f% 
of  its  cost.     Find  the  cost. 

107.  Five  men  in  a  factory  accomplish  as  much  as  8  boys. 
What  per  cent  of  a  man's  work  does  a  boy  do  ?     What  per  cent 
of  a  boy's  work  does  a  man  do  ? 

108.  Forty -five  per  cent  of  a  carload  of  melons  were  sold  to 
one  dealer,  and  33^%  of  those  left  to  another.     How  many  were 
there  in  the  car  before  any  were  sold,  if  after  the  second  sale 
there  remained  110  ? 

109.  In  a  certain  school  48%  of  the  pupils  are  boys,  and  there 
are  39  girls.     Find  the  number  of  boys. 

110.  How  many  Ib.  of  flour  will  be  required  to  make  1000  Ib. 
of  bread,  if  the  bread  weigh  30%  more  than  the  flour  used  ? 

111.  93  Ib.  6  oz.  is  what  per  cent  of  43  Ib.  12  oz.  ? 

112.  Water,  in  freezing,  expands  10%.     If  1  cu.  ft.  of  water 
weighs  1000  oz.,  find  the  weight  of  1  cu.  ft.  of  ice. 

113.  Give  answers  to  the  following : 

(a)  15 f  %  of  660  =  (d)  .2%  of  40  = 

(6)  660  is  15f  %  of  what  number  ?    (e)  40  is  .2%  of  what  number  ? 

(c)   ^  is  what  per  cent  of  f  ? 

(/)  What  per  cent  of  itself  must  be  added  to  a  number  so  that 
the  sum  diminished  by  10%  of  itself  may  be  17%  more  than  the 
original  number  ? 

114.  Brooms  are  bought  wholesale  at  $  20  a  gross.     What  per 
cent  profit  will  be  made  by  selling  them  at  20^  each  ? 

115.  A  merchant  purchases  sugar  at  $4.50  per  cwt.     At  what 
price  per  Ib.  must  he  sell  it  in  order  to  gain  5j%? 

116.  I  bought  a  house  for  $4000  and  spent  40%  of  the  cost 
in  repairs.     What  must  I  rent  it  for  a  month  in  order  to  make 
a  clear  gain  of  5%  of  the  total  cost,  taxes  and  repairs  amounting 
to  $  72  yearly  ? 


334  ARITHMETIC 

117.  By  selling  a  piano  for  $260,  a  dealer  loses  20%.     How 
much  should  he  have  sold  it  for  to  gain  5%? 

118.  A  man  having  lost  20%  of  his  capital  is  worth  exactly 
as  much  as  another  who  has  just  gained   15%  on   his   capital. 
The  second  man's  capital  was  originally  $  9000.     What  was  the 
first  man's  capital  ? 

119.  A  dealer  sold  an  article  for   $8.10  and  lost  10%.     At 
what  selling  price  would  he  have  gained  10%? 

120.  A  bookseller  deducts  10%  from  the  market  price  of  his 
books,  and  after  this  has  a  gain  of  25%.     He  sells  a  book  for 
$  7.20.     Find  the  cost  price  of  the  book,  and  what  per  cent  the 
marked  price  is  in  advance  of  the  cost  price. 

121.  A  merchant  bought  1000  yd.  of  carpet  at  60^  a  yard,  and 
sold  |  of  it  at  a  profit  of  30%,  i-  at  a  profit  of  20%,  and  the  rest 
at  a  loss  of  20%.     How  much  did  he  receive  for  the  carpet  ? 

122.  A  sells  goods  to  B  at  a  gain  of  12%,  and  B  sells  the  same 
goods  to  C  at  a  gain  of  7±-%.     C  paid  $3762.50  for  the  goods. 
How  much  did  A  pay  for  them  ? 

123.  A  machinist  sold  two  seed-drills  for  equal  sums  of  money. 
He  gained  25%  on  the  one  and  lost  25%  on  the  other.     His  total 
loss  was  $  9.60.     Find  the  cost  of  each  drill. 

124.  B,  purchased  a  house  and  lot  for  $3300,  paid  $975  for 
repairs,  and  now  rents  the  premises   for  $  30  a  month.     If  he 
expends  annually  for  taxes   $  48.70,  and   for  incidental    repairs 
$  35,  what  is  his  per  cent  of  annual  income  on  his  investment  ? 

125.  A  merchant  closed  out  a  stock  of  cloaks  for  $311.04, 
at  a  loss  of  28%.     Required  the  loss  by  the  transaction. 

126.  By  selling  my  cloth  at  $  1.26  per  yd.  I  gain  11^  more  than 
I  lose  by  selling  it  at  $  1.05  per  yd.     What  would  I  gain  by  sell- 
ing 800  yd.  at  $  1.40  per  yd  ? 

127.  A  merchant  marks  his  goods  at  40%  in  advance  of  cost, 
and  in  selling  uses  a  Ib.  weight  1  oz.  too  light.     If  he  throws  off 
10%  of  his  marked  price,  find  his  gain  per  cent. 


MISCELLANEOUS   EXERCISE  335 

128.  State  the  relation  between  1  Ib.   Troy  and  1  Ib.  Avoir. 
What  is  the  gain  per  cent  when  the  selling  price  per  oz.  Avoir. 
is  the  same  as  the  cost  per  oz.  Troy  ? 

129.  A  man  bought  a  bankrupt  stock  at  60^  on  the  dollar  of 
the  invoice  price,  which  was  $4840.     He  sold  half  of  it  at  10% 
advance  on  invoice  price,  half  the  remainder  at  20%  below  invoice 
price,  and  the  balance  at  50%  of  invoice  price.     His  expenses 
were  10%  of  his  investment.     Find  his  loss  or  gain  (a)  in  money, 
and  (6)  in  rate  per  cent. 

130.  The  list  price  of  an  article  is  $150.     If  trade  discounts 
of  25%  and  16f  %  are  allowed,  what  is  the  net  price  ? 

131.  If  a  dealer  buys  stoves  at  a  discount  of  22%  from  list 
price,  and  sells  them  at  list  price,  what  is  his  per  cent  of  gross 
profit  on  the  investment  ? 

132.  Eequired  the  net  price  of  an  article  listed  at  $400,  30%, 
10%,  and  5%  off. 

133.  From  the  list  price  of  a  line  of  goods  a  purchaser  is 
allowed  a  trade  discount  of  20%  ;    a  further  discount  of  10% 
off  the  trade  price  for  taking  a  quantity,  and  a  still  further  dis- 
count of  5%  off  his  bill  for  cash.     Find  his  gain  per  cent  by  sell- 
ing at  10%  less  than  the  list  price. 

134.  The  net  price  of  a  reaper  is  $158.40,  and  the  trade  dis- 
counts allowed  are  20%  and  10%.     Find  the  list  price. 

135.  A  commission  merchant  sold  coffee  for  me  and  remitted 
$1960,   after  deducting  his   commission    of   2%.     What   is  the 
value  of  the  coffee  ? 

136.  If  an  agent  receives  $  1092  to  buy  pork,  how  many  Ib., 
at  6^  per  Ib.,  can  he  buy  and  retain  his  commission  of  5%  for 
buying  ? 

137.  A  commission   merchant    sold  1014   bu.  of  oats,  at  41^ 
per  bu.,  paid  $33.74  freight  charges,  and  retained  3^-%  commis- 
sion.    How  much  should  he  remit  to  the  consignor  ? 


336  ARITHMETIC 

138.  A  lad  earned  §21.16  collecting  accounts  for  a  physician. 
He  was  allowed  5f  %.     What  amount  did  he  collect  ? 

139.  Find  the  premium  paid  to  insure  a  house  worth  $7500 
for  §  of  its  value  for  3  yr.,  the  rate  being  f  %  of  the  policy  for 
each  year. 

140.  What  premium  must  be  paid  to  insure  a  cargo  of  4880  bu. 
of  wheat,  valued  at  78^  per  bu.,  at  1£%,  the  policy  being  for  only 
I-  of  its  value  ? 

o 

141.  A  building  is  insured  for  $  400  more  than  f  of  its  cost  at 
4%.     If  destroyed,  the  loss  will  be  $216.     Find  the  cost  of  the 
building. 

142.  A  dealer  shipped  200  bbl.  of  apples   to   Liverpool;   the 
average  cost  of  the  apples  was  $  3.75  per  bbl.     For  what  sum  must 
he  have  the  apples  insured  at  f  %  premium  to  guard  against  all 
loss,  in  case  of  shipwreck,  his  other  expenses  being  $  75  ? 

143.  If  in  a  certain  town  $3093.75  was  raised  from  a  f  %  tax, 
what  was  the  assessed  valuation  of  the  property  in  the  town  ? 

144.  A  tax  of  $24,750  is  levied  on  a  town,  the  assessed  valua- 
tion being  1.5  mills  on  a  dollar.     What  tax  does  a  man  pay  on  an 
income  of  $  1100,  of  which  $  400  is  exempted  ? 

145.  A  farmer  whose  property  is  assessed  at  $9600  pays  on 
the  dollar,  1J  mills  for  township  rates,  1±-  for  county  rates,  1J 
for  railway  bonus,  and  21  for  school  rate.     How  much  does  he 
pay  in  all  ? 

146.  B's  tax  was  $  86.2755  when  the  rate  was  7.635  mills  on  a 
dollar.     What  was  the  assessed  valuation  of  his  property? 

147.  A  certain  school  section  is  assessed  for  $150,000.     The 
trustees  have  built  a  schoolhouse  costing  $  1800. 

(a)  What  will  the  schoolhouse  cost  a  ratepayer  whose  property 
is  assessed  for  $  4500  ? 

(b)  What  would  be  the  rate   of  taxation  per  annum  on  the 
whole  section  if  the  house  were  paid  for  in  six  equal  annual  pay- 
ments, without  interest  ? 


MISCELLANEOUS   EXERCISE  337 

148.  A  clerk  pays  $  7.50  taxes   on  his   salary.     What  is  his 
total  salary  if  $  400  of  it  is  exempt  from  taxation  and  a  2|%  rate 
is  levied  on  the  remainder  ? 

149.  What  per  cent  must  be  assessed  on  $  1,500,000  to  produce 
$29,400  after  paying  2%  for  collecting? 

150.  An  importer  receives  an  invoice  of  kid  gloves  billed  at 
$680,  pays  a  duty  of   50%   ad  valorem,  and  sells  them  at  an 
advance  of  33^%  on  their  gross  cost  to  him.     How  does  the  price 
paid  by  the  purchaser  compare  with  the  exporter's  price  ? 

151.  A  merchant  imports  75  cases  of  indigo,  gross  weight  196 
Ib.  each,  allowing  15  %  for  tare.    What  was  the  duty  at  5p  per  Ib.  ? 


152.  What  will  $1  amount  to  in  3  yr.  216  da.,  at  1\%  per 
annum,  simple  interest  ? 

153.  Find   the   simple   interest   on   $597.50   for  2  yr.  5  mo. 
12  da.,  at  8%  per  annum. 

154.  How  long  will  it  take  $450,  at   8%,  to   yield   $21.30 
interest  ? 

155.  What  amount  will  be  due  July  1,  1896,  on  a  note  of  $80, 
drawn  Feb.  6,  1896,  and  bearing   interest  at  5J%   per  annum, 
exact  interest  ? 

156.  Find  the  sum  due  Sept.  2,  1893,  on  a  note  for  $  147.33, 
given  Jan.  13,  1893,  and  bearing  interest  at  4%  per  annum. 

157.  Find  the  exact  interest  on  $225  from  July  13,  1893,  to 
Sept.  3,  1893,  at  6%. 

158.  Find  the  interest  on  $  1,  at  "\%  per  annum,  from  Jan.  1, 
1895,  to  June  3,  1895.     (Complete  answer  required.) 

159.  What  sum  will  amount  to  $354.09  in  7  mo.,  at  3%  per 
annum  ? 

160.  In  what  time  will  $  1350  earn  $  31.88  at  5%  per  annum  ? 

161.  Find   the   face   of   a   draft   that   cost    $434.70,   at 
premium. 

z 


338  ARITHMETIC 

162.  If  the  interest  is  $12.57,  the  time  8  mo.  2  da.,  and  the 
rate  per  annum  5^%,  what  is  the  principal  ? 

163.  Find  the  exact  interest  on  $  150  from  July  16  to  Dec.  9, 
at  5%  per  annum. 

164.  A  person  borrows  money  for  6  yr.  at  3^%,  simple  interest, 
and  repays  at  the  end  of  the  time,  as  principal  and  interest,  $  847. 
How  much  did  he  borrow  ? 

165.  Find  the  simple  interest  on  $912.50,  at  8%,  from  Feb. 
13,  1895,  to  Dec.  19,  1896. 

166.  A  note  of  $360,  drawn  April  20,  1895,  is  paid  July  2, 
1896,  with  interest  at  1\%  per  annum.     Find  the  amount  paid, 
simple  interest. 

167.  Oct.   15,  1895,  a  young  man  deposited  in  the  savings 
bank  the  sum  of  $  860.75.     May  20,  1896,  he  withdrew  the  prin- 
cipal and  simple  interest  at  4%  per  annum,     What  amount  did 
he  withdraw  ? 

168.  Bought  a  horse  for  $160,  and  gave  in  payment  my  note 
dated  Aug.  15,  1896,  with  interest  at  1\%  per  annum  until  paid. 
Jan.  9,  1897,  I  sold  the  horse  for  $  200  cash,  and  paid  my  note. 
What  was  my  net  gain  ? 

169.  If  for  $  7  I  can  have  the  use  of  $  35  for  3  yr.  4  mo.,  how 
much  a  month  shall  I  have  to  pay  for  the  use  of  $  8750  ? 

170.  Jan.    1,    1894,    a  person   borrowed    $2417.50    at   6|%, 
simple  interest,  promising  to  return  it  as  soon  as  it  amounted 
to  $  2582.50.      On  what  day  did  the  loan  expire  ?     (365  da.  = 


171.  March  1,  1896,  a  storekeeper  bought  goods  amounting, 
at  catalogue  prices,  to  $840,  on  which  he  was  allowed  succes- 
sive discounts  of  33^%  and  5%.  The  account  is  payable  in 
60  da.,  after  which  time  interest  is  to  be  charged  at  7%  per 
annum.  June  1,  1896,  he  paid  $  100.  How  much  is  due  July  1, 
1896? 


MISCELLANEOUS   EXERCISE  339 

172.  Find  the  proceeds  of  a  note  for  $200  given  at  Albany, 
N.Y.,  for  3  mo.,  and  discounted  at  bank  the  day  it  was  made  at 
6%. 

173.  Find  the  proceeds  of  a  note  for  $168  due  Oct.  20,  1896, 
and  discounted  Sept.  25,  1896,  at  a  Brooklyn,  N.Y.,  bank,  at  6% 
per  annum. 

174.  $1234^.  ST.   Louis,  Jan.  15,  1894. 
Ninety  days  after  date,  I  promise  to  pay  A.  Bee,  or  order,  the 

sum  of  one  thousand  two  hundred  and  thirty -four  ^ffe  dollars,  at 

the  Bank  of  Commerce  here.     Value  received. 

C.  DEE. 

This  note  was  discounted  Feb.  10,  1894,  at  6%  per  annum. 
Find  the  proceeds. 

175.  A  note  for  $230,  drawn  Jan.  2,  1896,  at  3  mo.,  and  bear- 
ing interest  at  8%  per  annum,  is  discounted  Feb.  1  at  7%.     Find 
the  proceeds. 

176.  Find  the  proceeds  of  the  following  note  : 

$  2400.  HAMILTON,  OHIO,  Feb.  3,  1896. 

Five  months  after  date,  value  received,  I  promise  to  pay 
Thomas  Cowan,  or  order,  the  sum  of  two  thousand  four  -hundred 
dollars,  at  the  Bank  of  Hamilton,  with  interest  at  6%  per  annum. 

VANCE  ALLEX. 

Discounted  May  22,  1896,  at  1%. 

177.  The  discount  on  a  note  for  $  3600,  which  matured  April 
21,  1896,  and  was  discounted  Feb.  24,  1896,  was  $  45.60.     Find 
the  rate  of  discount. 

178.  A  buys  600  yd.  of  silk  at  95^  per  yd.,  and  sells  it  at  once, 
receiving  in  payment  a  90-day  note  for  $700,  which  he  at  once 
discounts  at  a  bank  at  6%  per  annum.     Find  the  gain. 

179.  For  what  sum  must  a  note  be  drawn  June  1,  1896,  pay- 
able in  90  da.,  so  that  when  discounted  June  14,  at  8%,  the  pro- 
ceeds will  be  $  717.20  ? 


340  ARITHMETIC 

180.  What  rate  of  interest  is  made  by  a  bank  which,  discounts 
a  90-day  note  at  6%  per  annum  ? 

181.  Jan.  1,  A  owes  a  bank  $15,000.     He  offers  for  discount 
certain  notes:   $2500  due  Feb.  15,  $3700  due  March  13,  and 
$7500  due  April  1.     If  these  are  discounted  at  8%  per  annum, 
how  much  cash  must  he  pay  ? 

182.  What  must  be  the  face  of  a  note  so  that  when  discounted 
at  a  bank  for  90  da.  at  6%,  the  proceeds  will  be  $  1969  ? 

183.  What  is  the  present  worth  of  a  note  for  $  540,  due  in  90 
da.,  drawing  interest  at  6%,  discounted  at  8%,  true  discount? 

184.  Find  the  proceeds  of  a  note  for  $  292.73,  discounted  at 
bank,  for  35  da.,  at  6%  per  annum,  exact  interest  method. 


185.  Find  the  value  of  (1.03)4. 

186.  A  man  has  the  choice  of  loaning  his  money  at  7|%,  com- 
pound interest,  or  at  8%,  simple  interest,  money  and  interest  to 
be  paid  at  end  of  3  yr.     Show  which  is  the  better  investment. 

187.  An  annual  deposit  of  $  250  is  made  with  a  loan  company 
which  pays  4%  per  annum  on  deposits,  compounded  half-yearly. 
Find  the  amount  of  all  these  deposits  when  the  fourth  has  been 
made. 

188.  June  30,   1890,  I  borrow  $16.50,  to  be  returned  April 
30,  1892.     With  compound  interest  at  6|%,  what  amount  must 
I  then  pay  ? 

189.  A  man  puts  $350  in  a  savings  bank  each  year,  making 
his  first  deposit  Dec.  31,  1893.     How  much  will  there  be  to  his 
credit  Jan.  1,  1897,  the  bank  adding  4%  per  annum  ? 

190.  A  owes  B  $400  due  in  1  yr.,  $300  due  in  2  yr.,  $200 
due  in  3  yr.     What  sum  paid  now  would  cancel  the  debt,  money 
being  worth  5%  per  annum,  compound  interest? 


MISCELLANEOUS   EXERCISE  341 

191.  A  lent  a  sum  of  money  for  2  yr.,  at  10  %   per  annum, 
interest  compounded  yearly.     B  lent  an  equal  sum  for  the  same 
time  at  10%   per  annum,  interest  compounded  half-yearly.     B 
gained  $  220.25  more  than  A.     Find  the  sum  each  lent. 

192.  A  man  rents  a  farm  for  two  years  at  $  441  per  annum,  the 
rent  for  any  year  being  supposed  to  be  paid  at  end  of  that  year. 
Money  being  worth  5%  per  annum,  compound  interest,  find  what 
sum  would,  in  advance,  pay  the  two  years'  rent. 


193.  I  invest  $39,900  in  6  per  cents  at  95.     What  is  my 
income  ? 

(a)  What  sum  invested  in  8%  bonds  at  33 \%  premium  will 
yield  an  income  of  $  1200  ? 

(6)  What  if  the  bonds  were  at  33^%  discount '.' 

194.  Find  the  cost  of  64  shares  of  railroad  stock  at  107 J, 
brokerage  J%. 

195.  Kequired  the  gain  011  28  shares  of  stock  bought  at  97^ 
and  sold  at  103^. 

196.  April  4,  1896,  20  shares  of  Chicago  City  Kail  way  stock, 
quoted  at  216,  were  sold,  brokerage  J-%.     Find  what  was  received 
by  the  owner  of  the  stock. 

197.  A  railroad  company  declared  a  dividend  of  1J%  for  the 
quarter  ending  Sept.  30,  1894.     If  the  stock  was  quoted  at  105, 
what  was  the  rate  of  income  per  annum  on  an  investment  in  the 
stock  of  that  company  ? 

198.  How  much  must  be  invested  in  U.  S.  5's,  at  1131,  to  secure 
an  annual  income  of  $  175  ? 

199.  A  man  invests  $12,000  in  3%  stock  at  75.     He  sells  out 
at  80  and  invests  1  of  the  proceeds  in  3|%  stock  at  96,  and  the 
remainder  at  5%  par.     Find  the  change  in  his  income. 

200.  A  man  owned  $8940  bank  stock,  which   paid  a  yearly 
dividend  of  41%.     He  sold  out  102f,  and  invested  the  proceeds 


342  ARITHMETIC 

in  Michigan  Central  stock  at  74f,  paying  a  yearly  dividend  of 
3%.  By  how  much  was  his  yearly  income  changed  by  the 
transfer  ? 

201.  What  must  be  the  market  value  of  6%   stock,  so  that 
after  paying  an  income  tax  of  16   mills  on  the  dollar,  it  may 
yield  5%  on  the  investment  ? 

202.  A  person  bought  stock  at  951,  and  after  receiving  a  half- 
yearly  dividend  of  7%   per  annum,  sold  out  at  92-1,  brokerage 
each  way  being  \°/0-     If  his  net  gain  was  $25,  how  much  stock 
did  he  buy  ? 

203.  If  a  5%  stock  sells  at  105,  how  much  must  be  invested 
in  it  to  yield  a  yearly  income  of  $  794,  after  paying  an  income 
tax  of  15  mills  on  the  dollar,  $  400  of  income  being  exempted 
from  taxation  ? 

204.  A  man 'in  vests  $  6000  in  5%  stock  at  120.     At  the  end  of 
1  yr.,  having  just  received  the  yearly  dividend,  he  sells  at  1211. 
How  much  better  off  is  he  than  if  he  had  loaned  his  money  at 
5%  per  annum  ? 

205.  I  own  $6000  of  bank  stock,  paying  an  annual  dividend 
of  5%.     How  much  will  my  annual  revenue  from  the  bank  stock 
be  reduced  by  selling  enough  of  it  at  72  to  pay  a  note  of  $  3735 
9  mo.  before  it  is  due,  reckoning  true  discount  at  5%  per  annum  ? 

206.  A  man  received  $495  as  dividend  on  his  bank  stock. 
He  sold  40  shares  ($100)  at  1431,  and  the  remainder  at  144-1, 
paying  1%  brokerage  on  each  transaction.     What  were  the  net 
proceeds  of  the  sales  ? 


207.  $  1200  is  to  be  divided  between  two  persons,  A  and  B,  so 
that  A's  share  is  to  B's  share  as  2  to  7. 

208.  What  is  the  ratio  of  3J  to  |  ?     Answer  in  per  cent. 

209.  Divide  1026  into  four  parts  that  shall  be  in  the  ratio 
of  3,  11,  17,  and  23. 


MISCELLANEOUS   EXERCISE  343 

210.  An  upright  pole  16  ft.  long  casts  a  shadow  5  ft.  4  in. 
long,  and  at  the  same  hour  the   shadow  of  a  tree  is   found  to 
be  26  ft.  9  in.     Required  the  height  of  the  tree. 

211.  The   sum  of  three   numbers  is  940.      The   first  number 
equals  |  of  the  second,  and  the  second  equals  T7^  of  the  third. 
Find  the  numbers. 


212.  If  18  men  do  f  of  a  piece  of  work  in  30  da.  of  10  hr.,  in 
what  time  should  15  men  do  the  whole  work,  working  9  hr.  a  da.  ? 

213.  If  10  yd.  of  muslin,  1^  yd.  wide,  cost  $1.30,  what  is  the 
cost  of  12  yd.,  1|  yd.  wide  ? 


214.  One-sixth  of  the  square  of  a  certain  number  is  384.    Find 
the  number. 

215.  Find  the  square  root  of  .6  correct  to  three  decimal  places. 


216.  Find,  within  one  inch,  the  side  of  a  square  whose  area  is 
5  A. 

217.  A  rectangular  field  whose  length  is  f  of  its  width  contains 

2  A.  112  sq.  rd.     Find  the  length  of  a  diagonal. 

218.  Required  the  base  of  a  right-angled  triangle  whose  hypote- 
nuse is  16J  ft.,  and  perpendicular  9|  ft. 

219.  A  ladder  78  ft.  long    stands    perpendicularly  against  a 
building.     How  far  must  it  be  pulled  out  at  the  foot  that  the  top 
may  be  lowered  6  ft.  ? 

u 

220.  A  road  runs   round  a  circular  pond;    the  outer  circum- 
ference is  440  yd.,  and  the  width  of  the  road  is  20  yd.     Find  the 
area  of  the  pond. 

221.  In  order  to  drain  a  swamp  a  ditch  was  dug  1  mi.  long, 

3  ft.  deep,  6  ft.  wide,  at  the  surface,  and  4  ft.  wide  at  the  bottom. 
Find  the  total  cost  at  9^  per  cu.  yd. 


344  ARITHMETIC 

222.  How  many  gal.  in  a  circular  cistern  6  ft.  in  diameter  and 
7  ft.  deep  ? 

223.  The  surface  of  a  cube  is  432  sq.  ft.     What  is  its  volume  ? 

224.  (a)  A  circular  cistern,  8  ft.  in  diameter  and  9  ft.  in  depth, 
is  filled  with  water  to  the  height  of  6  ft.      How  many  gal.  of 
water  in  the  cistern  ?     (1  cu.  ft.  =  7.48  gal.) 

(b)  If  a  sphere  whose  diameter  is  4  ft.  is  submerged  in  the 
water  in  the  cistern,  how  high  will  it  cause  the  water  to  rise  ? 

225.  How  many  cd.  are  there  in  a  cylindrical  log  20  ft.  long 
and  3  ft.  6  in.  in  diameter  ? 

226.  Find  the  diameter  of  a  circle  whose  area  is  equal  to  the 
sum  of  the  areas  of  two  circles  whose  diameters  are  12  in.  and 
16  in.  respectively. 

227.  Find  the  area  of  the  curved  surface  of  a  right  circular 
cone  the  radius  of  whose  base  is  3.5  in.,  and  whose  altitude  is 
7  in. 

228.  A  chord  of  a  circle,  whose  radius  is  12  in.,  subtends  a 
right  angle  at  the  centre  of  the  circle.     Find  the  area  of  the 
smaller  segment  cut  off  by  this  chord. 

229.  A  spherical  shell,  internal  diameter  14  in.,  is  filled  with 
water.     Its  contents  are  poured  into  a  cylindrical  vessel  whose 
internal  radius  is  14  in.       Find  the  depth  of  the  water  in  the 
cylinder. 

230.  The  sides  of  a  triangle  are  40,  45,  and  50  ft.  respectively. 
Find  the  length  of  the  perpendicular  from  the  vertex  to  the  side 
45  ft. 

231.  The  diameter  of  a  circular  plate  of  lead  is  13  in.     From 
this  is  cut  out  a  circular  plate  of  radius  6  in.,  and  the  remainder 
of  the  lead  is  moulded  into  the  form  of  a  circular  plate,  with  J  of 
the  former  thickness.     Find  the  diameter  of  this  plate. 

232.  The  sides  of  a  triangle  are  13,  14,  and  15  ft.     Find  its 
area  and  the  length  of  the  three  perpendiculars  from  the  angles 
on  the  opposite  sides. 


MISCELLANEOUS   EXERCISE  345 

233.  The  external  dimensions   of  a  rectangular  covered  box, 
made  of  inch  stuff,  are  7,  8,  and  9  ft.     Find  the  capacity  of  the 
box  and  the  quantity  of  lumber  in  it. 

234.  A  ball  of  yarn  3  in.  in  diameter  makes  one  mitten.     How 
many  similar  mittens  will  a  ball  6  in.  in  diameter  make  ? 


235.  A  farmer  employs  a  number  of  men  and  8  boys ;  he  pays 
the  boys  $  .65  and  the  men  $  1.10  per  day.     The  amount  that  he 
paid  to  all  was  as  much  as  if  each  had  received  $  .92  per  day. 
How  many  men  were  employed  ? 

236.  Two  men  start  from  the  same  point  at  the  same  time  to 
walk  in  the  same  direction  around  a  block  of  land  1^  mi.  on  each 
side.     A  goes  at  the  rate  of  4  mi.  and  B  3  mi.  an  hr.     Howr  far 
will  A  walk  before  he  overtakes  B  ? 

237.  A  commission  merchant  sells  a  consignment  of  wheat  for 
$27,500,  on  a  commission  of  2|%.     He  pays  $250  for  freight 
and  storage,  and  with  the  net  proceeds  buys  pork  at  $  6.25  per 
cwt.,  charging  21%  for  buying.     How  many  cwt.  of  pork  does  he 
buy,  and  what  is  the  amount  of  his  two  commissions  ? 

238.  Find  the  cost  of  the  material  required  to  fence  21  mi.  of 
railway  (both  sides),  posts  placed  8  ft.  apart,  an  8-in.  base  1  in. 
thick,  a  2  x  4  in.  rail  at  top,  and  6  strands  of  wire.     The  posts 
cost  121^  each,  the  lumber  $  14  per  M.,  and  the  wire  4^  per  Ib. 
(A  Ib.  of  wire  stretches  1  rd.) 

239.  A  number  of  two  digits  is  multiplied  by  3,  and  the  product 
placed  to  the  left  of  the  original  number.     Show  that  the  number 
so  formed  is  always  exactly  divisible  by  7. 

240.  A  merchant  reduces  the  marked  price  of  an  article  by 
a  certain  per  cent.     He  gives  the  same  per  cent  off  this  reduced 
price  for  cash.     The  cash  price  is  now  ||  of  the  original  marked 
price.     Find  the  rate  per  cent. 


346  ARITHMETIC 

241.  Divide  $916  among  A,  B,  and  C  so  that  5%  of  A's  share 
may  equal  1\%  of  B's,  and  121%  of  B's  may  equal  20%  of  C's, 

242.  A  starts  to  walk  from  P  to  Q  at  the  rate  of  4  mi.  an  hr., 
and  1  hr.  afterwards  B  starts  from  P  and  overtakes  A  in  4  hr. 
Walking  on,  B  arrives  at  Q  2  hr.  before  A.     Find  the  distance 
from  P  to  Q. 

243.  A  number  is   divisible  by  9  if  the  sum  of  its  digits  is 
divisible  by  9.     Why? 

244.  At  what  two  times  between  3  and  4  o'clock  are  the  hands 
of  a  watch  equally  distant  from  the  figure  III  ? 


NEW   AMERICAN   EDITION   OF 

HALL  AND  KNIGHT'S  ALGEBRA, 

FOR  COLLEGES  AND  SCHOOLS. 

Revised  and  Enlarged  for  the  Use  of  American  Schools 

and  Colleges. 

By  FRANK   L.  SEVENOAK,  A.M., 

Assistant  Principal  of  the  Academic  Department,  Stevens 
Institute  of  Technology. 

Half  leather.     12mo.    $1.1O. 

JAMES  LEE  LOVE,  Instructor  of  Mathematics,  Harvard  University, 
Cambridge,  Mass.:  —  Professor  Sevenoak's  revision  of  the  Elementary  Algebra 
is  an  excellent  book.  I  wish  I  could  persuade  all  the  teachers  fitting  boys  for  the 
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VICTOR  C.  ALDERSON,  Professor  of  Mathematics,  Armour  Institute, 
Chicago,  111.:  —  We  have  used  the  English  Edition  for  the  past  two  years  in  our 
Scientific  Academy.  The  new  edition  is  superior  to  the  old,  and  we  shall  certainly 
use  it.  In  my  opinion  it  is  the  best  of  all  the  elementary  algebras. 


AMERICAN   EDITION   OF 

ALGEBRA  FOR  BEGINNERS. 

By  H.  S.  HALL,  M.A.,  and  S.  R.  KNIGHT. 

REVISED    BY 

FRANK  L.   SEVENOAK,  A.M., 

Assistant  Principal  of  the  Academic  Department,  Stevens 
Institute  vf  Technology. 

16mo.    Cloth.    6O  cents. 

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JAMES  S.  LEWIS,  Principal  University  School,  Tacoma,  Wash  :  —  I  have 
examined  Hall  and  Knight's  "Algebra  for  Beginners"  as  revised  by  Professor  Sev- 
enoak,  and  consider  it  altogether  the  best  book  for  the  purpose  intended  that  I 
know  of. 

MARY  McCLUN,  Principal  Clay  School,  Fort  Wayne,  Indiana:  — I  have 
examined  the  Algebra  quite  carefully,  and  I  find  it  the  best  I  have  ever  seen  Its 
greatest  value  is  found  in  the  simple  and  clear  language  in  which  all  its  definitions 
are  expressed,  and  in  the  fact  that  each  new  step  is  so  carefully  explained.  The  ex- 
amples in  each  chapter  are  well  selected.  I  wish  all  teachers  who  teach  Algebra 
might  be  able  to  use  the  "Algebra  for  Beginners." 


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HALL  and  KNIGHT'S 
ELEMENTARY  TRIGONOMETRY, 

WITH  TABLES. 

By  H.  S.  HALL,  M.A.,  and  S.  R.  KNIGHT,  B.A. 

Revised  and  Enlarged  for  the  Use  of  American 
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ELEMENTARY  SOLID  GEOMETRY. 

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i6mo.     Cloth.     $1.10,  net. 


PROF.  JOHN  F.  DOWNEY,  University  of  Minnesota  :-- There  is  a  gain  in 
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treatment  are  such  as  to  develop  in  the  student  ability  to  do  geometrical  work.  The 
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more  English  and  American  editions  of  Geometry  which  I  have  on  my  shelves,  I 
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By  J.  B.  LOCK, 

Author  of '"  Trigonometry  for  Beginners"  "Elementary   Trigonometry"  etc 

Edited  and  Arranged  for  American  Schools 

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process,  however  simple,  is  deemed  unworthy  of  clear  explanation.  Where  it  seems 
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